Nonlinear Optics in Semiconductors I

Nonlinear Optics in Semiconductors I SEMICONDUCTORS AND SEMIMETALS Volume 58 Semiconductors and Semimetals A Treatise...

Author: Robert K. Willardson | Dean E. Garmire Elsa Garmire

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Nonlinear Optics in Semiconductors I SEMICONDUCTORS AND SEMIMETALS Volume 58

Semiconductors and Semimetals A Treatise

Edited by R K Willarhon

Eicke R Weber

CONSULTING PHYSICISTDEPARTMENT OF MATERIALS SCIENCE SPOKANE, WASHINGTONAND MINERALENGINEERING UNIVERSITY OF CALIFORNIA AT BERKELEY

Nonlinear Optics in Semiconductors I SEMICONDUCTORS AND SEMIMETALS Volume 58 Volume Editors

ELSA GARMIRE THAYER SCHOOL OF ENGINEERING DARTMOUTH COLLEGE HANOVER. NEW HAMPSHIRE

ALAN KOST HUGHES RESEARCH LABORATORIES MALIBU. CALIFORNIA

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Contents

PREFACE. . . . . . . LISTOF CONTRIBUTOR^ .

.......................... ..........................

xi xv

Chapter 1 Resonant Optical Nonlinearities in Semiconductors Alan Kost I. INTRODUCllON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. SURVEY OF NONLINEAROpncAL MECHANISMS. ................ 1. State Filling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Coulomb Screening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bandgap Renormalization. . . . . . . . . . . . . . . . . . . . . . . . . . 4. Broadening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Quantum Wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Other Mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. MODELINGAND MEASURINGOPTICAL NONLINEARITY. ............. 1. SimpIeModels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Btinyai-Koch Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Kramers-Kronig Relation. . . . . . . . . . . . . . . . . . . . . . . . 4. Nonlinear Transmissionand Its Relation to Nonlinear Absorption and Refraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. RESONANTOPTICAL NONLINEARITY IN GaAs QUANTUM WELLS. . . . . . . . . 1. Sample Design and Fabrication . . . . . . . . . . . . . . . . . . . . . . . 2. Linear Optical Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Density-Dependent Absorption and Refractive Index . . . . . . . . . . . . . 4. Intensity-Dependent Absorption . . . . . . . . . . . . . . . . . . . . . . . V. SUMMARY OF BAND-FILLINGNONLINEARITIES .................. 1. Bulk Semiconductorsand Quantum Wells . . . . . . . . . . . . . . . . . . 2. Intersubband Absorption Saturation . . . . . . . . . . . . . . . . . . . . . 3. Quantum Dots and Semiconductor Doped Glosses . . . . . . . . . . . . . . 4. Optical Modulators and Active Media . . . . . . . . . . . . . . . . . . . . VI. FIGURESOPMERIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. OFTICAL NONLINEARITY FROM FRFECARRIER ABSORPTIONAND REFRACTION. . 1. BasicEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V

2 3 3 6 6 6 7 1 8 8 12 12 14 16 16 18

19 25 29 29 31 32 32 34 38 38

vi

CONTENTS

2. Nonlinear Optical Susceptibilities . . . . . . . . . . . . . . . . . . . . . . 3. Optical Switching of Microwaves . . . . . . . . . . . . . . . . . . . . . . . VI11. OFTOTHERMAL OPTICAL NONLINEARITIES . . . . . . . . . . . . . . . . . . . . IX . ALL-OPTICAL SWITCHING ............................ 1 . The Optical Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Nonlinear Fabry-Perot . . . . . . . . . . . . . . . . . . . . . . . . . 3. Demonstrations of Optical Bistahi1it.v and Optical Logic . . . . . . . . . . . X . SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF AEBREVlAnONS AND ACRONYMS . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 41

44 45 45 45 47 49 49 50

Chapter 2 Optical Nonlinearities in Semiconductors Enhanced by Carrier Transport Elsa Garmire LIST OF ACRONYMS. . . . . . . . . . . . . . . . . . . . . . . I . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Locul Nonlinearities Enhanced by Carrier Transport . . . . . . . . . . . 2 . Nonlocal Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . 3. Figures of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 11. EXPERIMENTAL RESULTS ON OPTICAL NONLINEARITIES INFLUENCED BY CARRIER TRANSPORT . . . . . . . . . . . . . . . . . . . . . . . . . 1. Enhanced Nonlinearities Based on State Filling with Decreased Carrier Recombination Rates . . . . . . . . . . . . . . . . . . . . . . . . 2 . Enhanced Nonlinearities Based on Photomodulationof Internul Fieluk . . . 3 . Combined Nonlinearities . . . . . . . . . . . . . . . . . . . . . . 4 . Self-Modulation of E.rternal Fields . . . . . . . . . . . . . . . . . . 111. FIELDDEPENDENCE OF THE OPTICAL hOPERTIES OF SEMICONDUCTORS . . . . 1 . Absorption Spectra in Direct-Band Semiconductors . . . . . . . . . . . 2. From- Keldysh Efect . . . . . . . . . . . . . . . . . . . . . . . . 3 . Kramers-Kronig Relation: Electrorefruction . . . . . . . . . . . . . . 4 . Quunturn Confined Stark Effect . . . . . . . . . . . . . . . . . . . 5. Advanced Quantum Confined Stark Concepts . . . . . . . . . . . . . . 6. QCSE at Other Wavvlengths . . . . . . . . . . . . . . . . . . . . . I . Electrically Controlled State Filling . . . . . . . . . . . . . . . . . . 1V . EXPERIMENTAL CONFIGURATIONS . . . . . . . . . . . . . . . . . . . . 1 . Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . A bsorpiion-Only Interferometer . . . . . . . . . . . . . . . . . . . 3 . Interferometer Based on Phase Shvt . . . . . . . . . . . . . . . . . 4 . Fuhrv-Perot Geometries . . . . . . . . . . . . . . . . . . . . . . . 5 . Bragg Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Four- Wave Mi.ring . . . . . . . . . . . . . . . . . . . . . . . . . v . CHARACTERISTICSOF EXPERIMENTAL DEVICES THAT UTILIZE SELF-MODULATION . 1 . Speed of nipi Structures . . . . . . . . . . . . . . . . . . . . . . . 2. M & h g Type nipi Structures . . . . . . . . . . . . . . . . . . . . 3 . Picosecond Excitation of nipi’s . . . . . . . . . . . . . . . . . . . . 4 . Laterul Enhanced Diflirsion . . . . . . . . . . . . . . . . . . . . .

56 56 57

60 61 62 63

64 74 80 81 86 86 88

90 91 102 113 122

123 123 126 127 128 139 146 146 147 149 154 I55

vii

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5. Experimental Performance of nipi’s Inserted into Devices . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . .

156 160

1. INTRODUCTION ........................ . 11. NEAR-BAND-GAP EXCITATIONS . . . . . . . . . . . . . . . . . . . Ill . TIMESCALES AND DYNAMIC TRENDS. . . . . . . . . . . . . . . . . IV . A PURELY COHERENT PROCESS INVOLVING ONLYVIRTUALELECTRON-HOLE PAIRS: THEEXCITONICOFTICAL STARKEFFECT . . . . . . . . . . . V. FUNDAMENTALS OF TWGPARTICLE CORRELATION E m s INVOLVINGRFAL

175 118 188

.

Chapter 3 Ultrafast Transient Nonlinear Optical Processes in Semiconductors D. S. Chemla

ELECTRON-HOLE PAIRS . . . . . . . . . . . . . . . . . . . . . V I . APPLICATIONS:SPECTROSCOPY AND DYNAMICS OF ELECTRONIC STATESI N HETEROSTRUCTURES. . . . . . . . . . . . . . . . . . . . . . VII . FUNDAMENTALS OF FOUR-PARTICLE CORRELATION EFFECTS INVOLVING REAL ELECTRON-HOLE PAIRS . . . . . . . . . . . . . . . . . . . . . . V I I I. DYNAMICS IN THE QUANTUM Kimncs REGIME . . . . . . . . . . . . IX . CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . LISTOF ABBREVIATIONS AND ACRONYMS . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 Optical Nonlinearities in the Transparency Region of Bulk Semiconductors Mansoor Sheik-Bahae and Eric W. Van Stryland 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Nonlinear Absorption and Refraction . . . . . . . . . . . . . . . . . 2. Nonlinear Polarization and the Definitions of Nonlinear Coeflcients . . . .

. . . . . . . . 2. Nondegenerate Nonlinear Refraction . . . . . . . . . . . . . . . . . 3. Polarization Dependence and Anisotropy of x(’’ . . . . . . . . . . . . .

111. THFDRY OF BOUND-ELECTRONIC NONLINEARITIES: TWO-BANDMODEL 1. Nondegenerate Nonlinear Absorption . . . . . . . . . . . . .

I v. BOUND-ELECTRONIC OPTICAL NONLINEARITIES IN ACTIVESEMICONDUCTORS . V . FREE-CARRIER NONLINEARITIES ..................... V I . EXPERIMENTAL METHODS . . . . . . . . . . . . . . . . . . . . . . . 1. Transmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Beam Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Excite-Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . . 5. Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.Z-Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Excite-Probe Z-Scan . . . . . . . . . . . . . . . . . . . . . . . . 8. Femtosecond Continuum Probe . . . . . . . . . . . . . . . . . . . . 9. Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

198 206 218 226 239 248 250 250

259 259 259 259 271 271 277 283 284 287 293 294 294 296 291 299 300 302 306 307

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CONTENTS

VII . APPLICATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Ultrafast All-Opticul Switching Using Bound-Electronic Nonlinearities . . 2 . Optical Limiting . . . . . . . . . . . . . . . . . . . . . . . . . . VfII . CONCLUSlON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LISTOF ABBREVIATIONS AND ACRONYMS . . . . . . . . . . . . . . . .

. .

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

308 308 309 311 313 314

Chapter 5 Photorefractivity in Semiconductors James E. Millerd. Mehrdad Ziari and Afshin Partovi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Plane- Wave Interference Model . . . . . . . . . . . . . . . . . . . 2. Simpl@ed Band Transport Model . . . . . . . . . . . . . . . . . . . 3. Steady-State Solution . . . . . . . . . . . . . . . . . . . . . . . . 4. Lineur Electro-Optic Effect . . . . . . . . . . . . . . . . . . . . . 5. Other Dielectric Modulation Mechanisms . . . . . . . . . . . . . . . I l l . BEAMCOUPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Coupled-Wave Equations . . . . . . . . . . . . . . . . . . . . . . 2 . Spatial Frequency Dependence . . . . . . . . . . . . . . . . . . . . 3. Intensity Dependence . . . . . . . . . . . . . . . . . . . . . . . . 4. Temporal Response . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Electron-Hole and Multidefrct Interactions . . . . . . . . . . . . . . . IV. FOUR-WAVE MIXING. . . . . . . . . . . . . . . . . . . . . . . . . I . Degenerate Four- Wave Mixing . . . . . . . . . . . . . . . . . . . . 2 . Diffraction Eficiency Meusurements . . . . . . . . . . . . . . . . . . 3. Sev- Pumped Phase Conjugation . . . . . . . . . . . . . . . . . . . 4. Polurization Switching . . . . . . . . . . . . . . . . . . . . . . . v . ENHANCED WAVE-MIXING TECHNIQUES . . . . . . . . . . . . . . . . . 1. DCAppliedFields . . . . . . . . . . . . . . . . . . . . . . . . . 2 . ACFirldr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Moving Graiings . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Temperature-in tens it^ Resonance . . . . . . . . . . . . . . . . . . . 5 . Neur-Band-Edge Effects . . . . . . . . . . . . . . . . . . . . . . . 6. Photorefractive Response at High Modulation Depths . . . . . . . . . . 7. Summary of Applied Field Techniques . . . . . . . . . . . . . . . . . V1 . BULKSEMICONDUCI'ORS . . . . . . . . . . . . . . . . . . . . . . . . 1.

lNTRODUCTION

11. SPACE-CHARGE GRATING FORMATION .

1.GaAs . . . . . 2.InP . . . . . . 3.GuP . . . . . . 4.CdTe . . . . . 5.ZnTe . . . . . 6.c'dS . . . . . . 7. Bulk 11- VI AIIOYS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WELLS . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . MULTIPLE QUANTUM . . . . . . I . MQW-PR Devices Using the Quantum Confined Stark Effect . . . . . . . 2 . Elimination of the Deposited Luyers and Substrate Removal in PR-MQ Ws . 3. MOW-PR Using the Frunz-Keldvsh Efect . . . . . . . . . . . . . . . V111. SELECTED APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . 1 . Coherent Signal Detection ( Ahptive Interferometer) . . . . . . . . . . 2 . Optical Image Proces.ring . . . . . . . . . . . . . . . . . . . . . .

320 321 321 323 324 326 327 328 328 330 332 333 336 337 338 340 342 344 346 346 348 350 352 354 361 363 364 365 366 361 367 367 369 369 371 385 380 383 385 381 387

CONTENTS

3 . Optical Correlators . . . . . . . . . . . 4. Real-Time Holographic Interferometry . LISTOF ABBREVIATIONS AND ACRONYMS. .

Remimas

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INDEX CONTENTS OF VOLUMES IN THIS SERIES

ix 388 391 394 395

403

409

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Preface This two-volume set is designed to bring together two streams of thoughtsemiconductors and nonlinear optics -and to bridge the gap between optics and electronics. Practical nonlinear optical devices in semiconductors are on the verge of becoming a reality, as switches, modulators, converters, and sensors. A high level of direct band semiconductor technology has come about because of the many practical applications of semiconductor lasers, modulators, and high-speed detectors. In general, their performance has been enhanced by using quantum confinement, particularly quantum wells. This technological base has provided high quality materials and devices from which nonlinear optical studies have proceeded. In particular, epitaxial films grown by technologies such as molecular beam epitaxy and metalo-organic chemical vapor deposition have extended the range of options available for optical nonlinearities. Nonlinear optics grew rapidly with the development of lasers, originating in the study of the interactions of laser light with dielectric media. Some early work was performed on semiconductors, but the high intensities initially needed for nonlinear optics required high power lasers and often caused permanent material damage. In the last twenty years, however, semiconductors have been shown to exhibit sensitive nonlinearities, which can be made particularly large by using quantum confinement or narrow gap materials. The steady increase in semiconductor materials technology has enabled a number of interesting applications for nonlinear optics and furthered the understanding of the basic physical principles taking place in both bulk and quantum confined systems. Optical nonlinearities in semiconductors can now be accessed by milliwatt lasers, e.g., laser diodes. Why publish a review now? We believe that at the present time most of the basic concepts in semiconductor nonlinear optics are well understood, and that a review of the basic science and technology in one place would be very useful. We have put together this two-volume set to stand together as xi

xii

PREFACE

a respresentation of the major thrusts of nonlinear optics in semiconductors. Most of the chapters contain enough basic background that they can be read without extensive additional study. Fundamental physical principles as well as engineering approaches and applications are presented. Most of the chapters have a balanced account of both experimental and theoretical advances. As often happens with books of this sort, some important areas were missed due to time and space constraints. We were not able to include nonlinear optics in semiconductor waveguides, microcavities, self-electrooptic effect devices, or to include fabrication technologies. Finally, we have chosen to concentrate on basic concepts more than applications, as these latter tend to become dated more rapidly. Volume 58 begins with a review of optical nonlinearities that arise from absorption-induced photocarriers and the effect of these electrons and holes on the absorption and refractive index near the band edge in semiconductors. Chapter one surveys local optical nonlinearities and models useful for device and applications designers. The resonant nonlinearities arise from screening and filling of available states in bulk and quantum wells, and include the role of excitons and free carriers. Methods for analyzing experimental measurements, figures of merit, and tables of published values are also presented. The second chapter investigates how the transport of photo-carriers affects resonant absorption and related refraction, through lengthening the lifetime of local nonlinearities and through non-local nonlinearities arising from photo-carrier screening fields. Examples of carrier transport nonlinearities, a study of electro-absorption in quantum wells, the properties of n-i-p-i structures, and characteristics of typical device configurations are all presented. in the third chapter Daniel Chemla reviews the status of the present understanding of the temporal evolution of optical nonlinearities in semiconductors. The availability of femtosecond lasers, along with phase measurements, have made possible studies of purely coherent processes involving only virtual transitions such as the optical Stark effect, as well as transitions involving real electron-hole pairs. This chapter introduces the many-body concepts necessary to interpret experimental data, separating out excitonic effects, as well as two-particle and four-particle correlation effects. Non-resonant optical nonlinearities occuring in the transparent region of semiconductors are considered in the last two chapters of the first volume. Sheik-Bahae and Van Stryland review two-photon absorption and its related nonlinear refraction, presenting a simple two-band model that fits a wide range of direct band semiconductor data. They review experimkntal methods for obtaining data and some applications in optical switching and limiting. Millerd, Ziari and Partovi review photorefractivity in semiconductors, a wave-mixing process that relates optically generated carriers and electro-

PREFACE

xiii

optical nonlinearities. Resonant nonlinearities provide enhanced performance in some cases, especially in photorefractive quantum wells. Advanced applications using photorefractivity are also outlined. Volume 59 continues the review of nonlinear optics in semiconductors with some of the newest research that promises an array of potential applications. Khurgin describes how the very large second-order nonlinearities in semiconductors can lead to harmonic generation, once advanced growth and fabrication techniques that enable phase-matching have been developed. Quantum wells and superlattices have large second order nonlinearities, particularly using intersubband transitions. Optical rectification and its resulting terahertz emission in semiconductors are also described. Hall, Thoen and Ippen explore optical nonlinearities in active semiconductor gain media, separating out diffusion, carrier scattering and carrier heating effects using femtosecond pump-probe techniques. Cross-phase and cross-gain modulation, spectral hole burning, and two photon absorption are all identified and applied to understanding semiconductor lasers and optical amplifiers. Shaping and saturation of pulses in active waveguides, four-wave mixing for wavelength conversion, and all-optical switching, all have application to broadband optical communications and switching systems. Hanamura provides a theoretical basis for the enhancement of the optical response in semiconductors due to quantum confinement. His analysis includes excitons and biexcitons in quantum wells, wires, and dots. The chapter provides experimental results and analysis on superradiance and coherent light emission as well as figures of merit for nonlinear optical response. Keller discusses passive switching of solid state lasers by using semiconductor saturable absorption integrated directly into a mirror structure. Various cavity designs are presented, including the anti-resonant FabryPerot and dispersive devices. Designs and results for mode-locked and Q-switched solid state lasers using these devices are included. Miller presents nonlinear optics using picosecond pulse measurements as a tool to better understand properties of semiconductor carrier transport. Determinations of carrier diffusion and mobility in bulk and quantum wells are made, separately considering lateral carrier motion in the wells and movement across the wells. The purpose of this two-volume set is to review research into the nonlinear optics of semiconductors, which has led both to a much better understanding of semiconductors and to the demonstration of various applications, from phase conjugation to optical switching to ultrashort pulse generation to optical limiting. While semiconductor nonlinear optical devices have not yet reached the level of importance that semiconductor lasers

xiv

PREFACE

and modulators have, a number of the semiconductor nonlinearities are important to the laser community and some have the potential to achieve wide deployment in applications. It is our hope that this book will serve those in the semiconductor community interested in nonlinear optics, those in nonlinear optics interested in semiconductors, and also as a general resource. ELSA GARMIRE ALAN KOST

List of Contributors

Numbers in parenthesis indicate the pages on which the authors’ contribution begins.

DANIEL S. CHEMLA, (175) Department of Physics, University of California at Berkeley; Director, Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California ELSA GARMIRE, (55) Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire ALANKOST, (1) Hughes Research Laboratories, Malibu, California

JAMES E. MILLERD, (319) MetroLaser, Inc., Irvine, California AFSHINPARTOVI, (319) Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey MANSOOR SHEIK-BAHAE, (257) Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico ERICW. VANSTRYLAND, (257) Center for Research and Education in Optics and Lasers (CREOL), University of Central Florida, Orlando, Florida MEHRDAD ZIARI,(319) SDL, Inc., San Jose, California

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SEMICONDUCTORSAND SEMIMETALS. VOL. sa

CHAPTER1

Resonant Optical Nonlinearities in Semiconductors Alan Kost HRL LABORATO~IES MALIBU. CALIFORNIA

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. SURVEY OF NONLINEAR OPTICAL MECHANISMS .................. 1. StateFilling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. CoulombScreening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bandgap Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 . Quantum Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. OtherMechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. MODELINGAND MEASURING OPTICAL NONLINEARITY ............... 1. SimpleModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Bhyai-Koch Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Kramers-Kronig Relation . . . . . . . . . . . . . . . . . . . . . . . . . 4. Nonlinear Transmissionand Its Relation to Nonlinear Absorption and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . R E ~ ~ N A NOPTICAL T NONLINEARITY IN GaAs QUANTUM WELLS. . . . . . . . . . 1 . Sample Design and Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 2. Linear Optical Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Density-Dependent Absorption and Refractive Index . . . . . . . . . . . . . . 4. Intensity-DependentAbsorption . . . . . . . . . . . . . . . . . . . . . . . . v . SUMMARY OF BAND-FILLING NONLINEARITIES ................... 1. Bulk Semiconductors and Quantum Wells . . . . . . . . . . . . . . . . . . . 2. Intersubband Absorption Saturation . . . . . . . . . . . . . . . . . . . . . . 3. Quantum Dots and Semiconductor Doped Glasses . . . . . . . . . . . . . . . 4. Optical Modulators and Active Media . . . . . . . . . . . . . . . . . . . . . VI . FIGURESOFMERIT ................................ VII . OFTICAL NONLINEARJTY FROM FREE CARRIER ABWRPTIONAND REFRACTION . . . 1. Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Nonlinear Optical Susceptibilities . . . . . . . . . . . . . . . . . . . . . . . 3. Optical Switching of Microwaves . . . . . . . . . . . . . . . . . . . . . . . VIII . OPTOTHERMAL OPTICAL NONLINEARITIES ..................... IX . ALL-OPTICAL SW~CH ING ............................. 1. The Optical Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Nonlinear Fabry-Perot . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Demonstrations of Optical Bisrability and Optical Logic . . . . . . . . . . . .

2 3 3 6 6 6 7 7 8 8 12 12 14 16 16 18 19 25 29 29 31 32 32 34 38 38 39 41 44 45 45

45 47

*List of Abbreviations and Acronyms can be located preceding the references to this chapter.

1 Copyright i t 1 1999 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752167-4 ISSN M)80-8784/99 530.00

2

ALAN Kos-r X. SUMMARYAND CONCLUSIONS ...........................

49

LISTOF ABBREVIATIONS AND ACRONYMS . . . . . . . . . . . . . . . . . . . . .

49

REFF.RENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Introduction When a semiconductor is exposed to light with photon energy above the absorption edge, it creates electron-hole pairs. A moderate density of excess particles significantly alters the optical properties of the semiconductor within a few hundred millielectronvolts of the absorption edge. In the late 197Os,it was shown that these effects were large enough for optical bistability in semiconductor etalons (Gibbs et al., 1979; Miller et al., 1979). This stimulated tremendous interest in resonant semiconductor optical nonlinearities, which are functions of the density of photogenerated electron-hole pairs and depend only indirectly on optical intensity. Resonant effects can be observed with a single optical beam or through the interaction of pump and probe beams, which need not be at the same wavelength or coincident in time. Many of the first studies of heavily doped and photoexcited semiconductors were made with small-bandgap materials (e.g., InSb), where the biggest effect of excess charge carriers is a filling of the energy bands (Burstein, 1954; MOSS, 1954).As a result, semiconductoroptical nonlinearities due to particles optically excited across the bandgap have come to be called band-filling nonlinearities. These and other resonant mechanisms are surveyed in this section. Direct-bandgap semiconductors are of most interest for resonant optical nonlinearities because they have an abrupt absorption edge that is easily

PHOTONENERGY

FIG. 1. The absorption edge for a direct-bandgap semiconductor

1 REWNANT OPTICAL NONLINEARITIES IN SEMICONDUCTORS

3

modified with optical excitation. A typical absorption edge for a directbandgap semiconductor is illustrated in Fig. 1. The absorption can be thought of as having two components. The first is from transitions between single-particle states. In an idealized bulk semiconductor with parabolic energy bands, this contribution increases like ( E - EG)’/’. Coulomb attraction between photoinduced electrons and holes increases the absorption of the energy bands and gives resonances that correspond to the formation of bound excitons. These effects are larger in larger-bandgap semiconductors (e.g., GaAs, ZnSe), where electrons and holes have smaller effective masses and are more tightly coupled by the Coulomb interaction. Excitonic absorption resonances are especially pronounced in semiconductor quantum wells. Crystal imperfections, impurities, phonons, and carrier-carrier scattering all broaden the absorption spectrum and generate an absorption tail. 11. Survey of Nonlinear Optical Mechanisms 1. STATEFILLING

An interband transition between single-electronstatescan take place only if the first state is occupied and the final state is empty. As photoexcitation empties a valence band and fills the conduction band, transition blocking occurs that decreases the absorption coefficient (Fig. 2). This is called bandjlling or state filling and phase spucejlling when it refers to the blocking of excitonic absorption. a.

The Blocking Factor

Without Coulomb enhancement or broadening, the absorption coefficient near the absorption edge of a bulk direct-bandgap semiconductor is (Yariv,

W

WAVEVECTOR

BAND LIGHT-HOLE BAND

FIG. 2. Photoexcited electrons and holes fill energy bands and block absorption.

ALAN Kosr

4

1989):

a(ho) =

J5 t?’n$/’P; m2wns,nch3

( h o - EG)”’B(~W)

where m, is the reduced mass for the electron and heavy hole, and P , is the interband momentum matrix element. The contribution from light holes usually can be neglected. For 111-V semiconductors, it is at least 30 times smaller. B ( h o ) is a blocking factor that accounts for state filling:

m4 = (1 - f,CE,(k)l

- f,CEc(k)l)

(2)

where E,,(k) and Ec(k) are the energies of the valence band and conduction band states, respectively, and f, and f, are Fermi functions for holes in the valence band and electrons in the conduction band, respectively. The blocking factor is just the probability that there is no hole at E,(k) minus the probability that there is an electron at E,(k). For high optical excitation at ho,the blocking factor approaches zero. With electrical injection of carriers or optical excitation with a separate light source at an energy other than hw, the blocking factor can be negative and corresponds to optical gain. Electrons in the conduction band dominate the blocking effect. If the excitation is not too strong, Boltzmann statistics apply, and the change in occupation probability for states in a parabolic band is given by

where N is the density of extra particles, mefris their effective mass, and E(k) is energy measured relative to the bottom of the band (Miller, 1981a). Equation (3) shows that the occupation probability changes most for the lighter conduction band electrons. Coulomb attraction between electrons and holes complicates this picture. The Coulomb interaction leads to a series of discrete excitonic absorption resonances just below the bandgap, which, for a direct-bandgap semiconductor, have absorption strength (Butcher and Cotter, 1990):

where &,(O) i s the envelope wavefunction for the nth exciton state evaluated st r = 0:

1 RESONANT OPTICAL NONLINEARITIES IN SEMICONDUCTORS

5

E , is the bandgap, E , is the exciton binding energy: E,,

mre4 1 8 ~ , Z ~n2 ih~

= --

and cr is the relative permittivity for the semiconductor. The lowest energy, n = 1, exciton state is the most prominent in the absorption spectrum and the only one that can be observed as a distinct peak at room temperature. The Coulomb interaction also enhances the absorption from the energy bands (Eq. 1) by the factor I&'"(r = 0)l2, where (6" is the envelope function for unbound but correlated electron-hole pairs (Madelung, 1981):

with

YeY I@yho, r = 0)l = sinh y

and

The blocking factors in Eqs. (4) and (7) are now more complicated expressions for which Eq. (2) is an approximation. b. Excitons

-

The exciton represents a superposition of single-electron states with a range of k values of l/ag, where a r is the usual Bohr radius increased by the dielectric constant of the semiconductor: a?=-

47Ch2&,&, e2mr

For GaAs, a? is x120A, so the exciton is made from states from k = 0 to O . O 1 k l , corresponding to a range energies from the bottom of the

-

conduction band to 5 meV away. Photoinduced particles in this range contribute to exciton blocking. At low temperatures, unbound electrons and holes, clustered in the states close to k = 0, generally give stronger blocking

6

ALAN KOST

of the excitonic absorption. At room temperature, excitons fill phase space more effectively and give stronger blocking of the excitonic absorption (Schmitt-Rink er al., 1985).

2. COULOMB SCREENING Photoinduced electron-hole pairs screen the Coulomb interaction. The wavefunctions &,, and @‘“ spread because electrons and holes are not so tightly bound. Both exciton and energy-band absorption decrease as the values of I&,(0)12 and I&”(0)12 become smaller (Fig. 3). Screening is somewhat less important for nonlinear effects in lowerdimensional semiconductors (i.e., quantum wells, quantum wires, and quantum dots) (Haug and Schmitt-Rink, 1985; Schmitt-Rink et al., 1987).

3. BANDGAPRENORMALIZATION The Coulomb interaction also “renormalizes” the single-particle energies. The result is a shrinkage of the bandgap in the presence of photoinduced electrons and holes and a corresponding red shift of the absorption from the energy bands. This can give increasing absorption at wavelengths just below the absorption edge of the unexcited semiconductor. There is almost no shift in the position of the low-energy exciton absorption peak under photoexcitation because the decrease in the exciton binding energy almost exactly cancels the band-gap shift. 4.

BROADENING

In real semiconductors, excitonic absorption lines are not infinitely narrow. They are broad and overlap the absorption from the energy bands. Photoexcitation increases scattering rates and further broadens the excitons, reducing their peak absorption.

FIG. 3. Screening spreads thc exciton wavefunction and reduces the magnitude at r

= 0.

1 RESONANT OPTICAL NONLINEARITIES IN SEMICONDUCTORS

7

Broadening also gives the absorption edge a tail. Empirically, there are portions of the tail that have an exponential form (Urbach, 1953). In this energy range, there is a photon energy E' such that the absorption for energy less than E' has the form

-

where E,, an energy between 5 and 7 meV at room temperature, is called the Urbach parameter. This portion of the absorption spectrum is called the Urbach tail. There is a long history of attempts to explain the origins of the Urbach tail (Cohen et al., 1988). Nonlinear absorption in the absorption tail is a subtle combination of the effects from broadening, bandgap renormalization, state filling, Coulomb screening, and impurity absorption. In cold InSb, Miller observed rapid saturation of absorption in the tail, which was attributed to a filling of acceptor states and a corresponding blocking of transitions to the conduction band (Miller, 1982; Miller et al., 1981). Raj and colleagues found that photocarriers very effectively saturated the absorption in the tail just below the exciton in a GaAs multiple quantum well structure (Raj et al., 1992). It is generally observed that it is increasingly difficult to saturate absorption in more weakly absorbing portions of the absorption tail at longer wavelengths. Liebler and Haug have presented a theory that appears to explain this observation (Liebler and Haug, 1990). 5.

QUANTUM

WELLS

The most distinctive optical features of semiconductor quantum wells are the pronounced excitonic absorption resonances. The excitons have a larger absorption strength than in bulk semiconductors, and an increased binding energy puts the excitons farther from the band edge. Excitonic absorption saturates with low photocarrier density in both bulk and quantum well semiconductors, but the effect is more pronounced in quantum wells, and the absolute absorption change is larger. The book by Weisbuch and Vinter (Weisbuch and Vinter, 1991) provides an excellent overview of the many unique properties of semiconductor quantum wells.

6. OTHERMECHANISMS Free carrier absorption, free carrier refraction, and optothermal effects also give optical nonlinearity. They are examples of resonant effects that are

8

ALAN Kosr

not considered band filling nonlinearities. Collectively, band filling nonlinearities, and nonlinearity from free carrier absorption, and free carrier refraction are often called free carrier nonlinearities. Optical nonlinearity from the free carrier absorption or refraction is most significant at long wavelengths (2 2 10 pm), while optothermal nonlinearities can be large at any wavelength.

111. Modeling and Measuring Optical Nonlinearity 1. SIMPLEMODELS a.

Saturation Density and Saturation intensity

Despite the underlying complexity of resonant semiconductor optical nonlinearities, they can be modeled with simple expressions useful for device design. The nonlinearities are directly related to increasing photocarrier density, so the absorption is often modeled with the approximate form:

where E is the photon energy, N is the photoinduced electron-hole density, and N,!, the saturation density, is the value of N for which the absorption falls to one-half its initial value. Better fits are often obtained by adding an unsaturating term to Eq. (12).

When excitation is with short optical pulses, carrier density depends on optical fluence. Then, instead of Eq. (12) one may use

Alternatively, it may be useful to model the intensity dependence of the absorption using

where I is the optical intensity, and I,,, is the saturation intensity. An tinsaturating term is added to Eq. (14) or (15) if it gives a better fit.

1 RMNANT OPTICAL NONLINEARITIES IN SEMICONDUCTORS

9

Near exciton absorption peaks, the absorption has been modeled with two saturating terms. The first term, which saturates rapidly, represents the excitonic contribution to the absorption. The second term represents more slowly saturating background absorption from the energy bands.

The nonlinear refractive index is modeled using similar equations:

or

where ns is the largest refractive index change. When the laser spot has a nonuniform profile, or when absorption substantially attenuates the beam along the length of the sample, the modeling equations should be integrated over space for best fit to the experimentally measured quantities of power and energy. In practice, variations along the length of the sample are often ignored altogether. The spatial profile of the beam is taken into account by defining an effective intensity. For a Gaussian spot, it is common to use I,, I P/n(r ,p)*,where rlle2 is the l/e-squared radius for the intensity of the spot.

b. The Relation Between Photocarrier Density and Optical Intensity Photocarrier density is given by the rate equation

where z ( N ) is the density-dependent electron-hole recombination time, and D is the ambipolar diffusion coefficient. The third term on the right-hand side represents diffusion of electrons and holes out of the region of excitation. When free carriers in the bands are mostly photocarriers, the number of electrons equals the number of holes, and the recombination rate can be

10

ALAN KOST

approximated by a polynomial: 1 --A

dN)

+ BN + C N Z

The first term in Eq. (20) comes from nonradiative recombination through defect states in the bandgap. It is called Shockley-Read-Hall recombination and has this particularly simple form when empty defect levels capture electrons much more rapidly than holes or vice versa. The second term comes from radiative transitions and the third from Auger recombination. The recombination time is constant if the first term dominates, but depending on the semiconductor and the temperature, this may be true only for low excitation. Carrier lifetime is strongly affected by sample quality, and it can vary by more than an order of magnitude for different samples of the same semiconductor. Carrier lifetime is also difficult to measure, so uncertainty in its value is a common source of error for calculations relating photocarrier density with optical intensity. For continuous-wave (CW) or long-pulse excitation, the left-hand side of the rate equation can be set to zero. If the illumination is also spatially uniform, the diffusion term can be neglected. Then we find the useful relation

N = -Iat ho When a and T depend on N, as is often the case, Eq. (21) must be solved self-consistently. In this case, N is not proportional to I, and it might be objected that Eqs. (12) and (15) (or Eqs. 17 and 18) cannot both be correct. The answer is that both expressions are approximations that are used when they work. Equations (12) and (17) are more reliable because resonant optical nonlinearities are directly related to carrier density. If the laser spot has a nonuniform profile, an effective intensity is used with Eq. (21). If the laser spot is also small, the effective intensity should take into account photocarrier diffusion out of the spot. When photocarriers diffuse only laterally, the laser spot has a Gaussian profile, and the recombination time is approximately constant, a frequently used definition is P

c. Estimate of the Saturation Level for Exciton Absorption The exciton saturation density can be estimated with simple physical arguments (Chemla et al., 1984). Saturation occurs when the density of

1

RESONANTOPTICAL NONLINEARITIE~ IN SEMICONDUCTORS

11

photogenerated electrons plus holes is about one charge carrier per exciton volume 47ra2/3. This picture gives

and

Using typical values for GaAs, that is, a, = l20& ho = 1.43 eV, a z 7500 cm- l, and z = 10ns, gives N::t = 7 x 10l6 cm-' and = 200 W/cm2.

d. Estimate of the Saturation Level for the Energy Bands Optical nonlinearities at moderate to high excitation are most important for nonlinear optical devices, which require relatively large absolute changes in absorption or refractive index. The changes from exciton saturation at low levels of excitation are usually not enough. At moderate to high excitation, most of the required fluence or intensity goes to blocking the absorption of the energy bands. The conduction band states fill first. The density of states for electrons at energy E, measured from the bottom of a parabolic conduction band in a bulk semiconductor, is

Most photoinduced electrons will occupy states that are within a few k,T of the bottom of the band. Half the number of these states gives an estimate for the number of electrons required to reach the saturation level, so

and

Using typical values for GaAs, that is ho = 1.43 eV, meff= 0.067m0, and t = lOns, gives N,b,B,= 8.5 x l O " ~ m - ~ , and Z:,j = 7.5kW/cmz.

a x 2500cm-',

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ALAN K m

2. THEBANYAI-KOCH MODEL

For detailed modeling of band-filling nonlinearities, Eq. (1) which neglects excitons, gives reasonable results for bulk semiconductors with small bandgap (Poole and Garmire, 1985). Semiempirical models for quantum wells have been presented by Chemla et al. (1984) and Kaushick and Hagelstein (1994). Rigorous, many-body models generally require extensive numerical calculation (Haug, 1988); but the widely used Banyai-Koch model for bulk semiconductors (Banyai and Koch, 1986; Koch et af., 1988) is comparatively easy to use and shows excellent agreement with experiment (Lee et af., 1986a; Olbright et al., 1987; Peyghambarian et al., 1988). A simple blocking factor is used for both excitons and the energy bands: E'(ho) = B c o n ( h ~=) tanh

2k, T

(28)

where E,, and E,, are the quasi-Fermi levels for electrons and holes, respectively. Analytical expressions for the exciton wavefunctions are obtained by solving the effective mass equation using the HulthCn potential to approximate the screened interaction between electrons and holes. The shift in the absorption edge from band-gap renormalization is set equal to the change in the exciton binding energy. Broadening is included by convolving all spectral features with the line shape: 1 = aT cosh ( E / T )

in order to reproduce the exponential form of the absorption tail. The broadening parameter is E, + rlN, where E, is the Urbach parameter for the unexcited semiconductor, and is a density-dependent broadening. The Banyai-Koch approach has been extended to quantum wells by Pereira (1 995). 3. THEKRAMERS-KRONIG RELATION

The resonant changes in absorption and refractive index are related by a modified K ramers-K ronig relation: Att(E,

ch

N) = - P a

Aa(E', N ) dE El2 - E 2

jo

1 RESONANT OPTICAL NONLINEARKIES IN SEMICONDUCTORS 13

where the P indicates that the integral is a Cauchy principal value. As a rule of thumb, if the largest value of Aa is 1OOOcm- the largest value of An will be -0.01, but not at the same energy. It is often easier and more accurate to measure Aa and calculate An with the Kramers-Kronig relation than to measure An directly. To use the equation properly, all the Aa(E', N) must be measured at the same photocarrier density. The equation cannot be used to find An from a set of Aa measured at a constant intensity. The Kramers-Kronig relation shows that An is largest near the absorption edge, where Au is also large. This means that the design of devices that use resonant nonlinear refraction may involve a tradeoff between larger nonlinear refractive index changes and smaller background absorption. The Kramers-Kronig expression relates the modeling equations discussed earlier. If the saturation density for absorption does not depend on wavelength, substituting Aa given by Eq. (12) or (13) gives Eq. (17) for the saturating refractive index. The saturation densities for absorption and refraction are the same, and the maximum change in refractive index is given by

',

or

To evaluate the Kramers-Kronig relation, it is convenient to use linear interpolation to turn a discrete set of data for Aa into a continuous set (Fig. 4). Then the integration can be carried out analytically. An(E) =

ch -P

i=o n

ch n ;=I)

=-

1

hi 'I+'

m i E + bi dE' E 2 - E2

1

E;+1 - E2 bi Ei+1 - E Ei + E +--In1 2 E El+, + E E i - E E 2 - E?

where

m.=

Aai+l - Aai Ei+ 1 - Ei

and

bi = Aai - miEi

(34)

14

ALANKOST

Eo

El

El

4rl

E,

EM1

FIG.4. A discrete data set for the absorption change is transformed into a continuous set by linear interpolation.

4. NONLINEAR TRANSMIWON AND ITS RELATIONTO NONLINEAR ABSORPTION AND REFRACTION The nonlinear absorption Aa = a - a(0) is not measured directly. A transmission change is measured instead. The fractional change ATIT is defined by

AT - T - T(0) -T - T(0)

(35)

where T is the photoexcited transmission, and T(0) is the initial transmission. Let us see how ATIT is related to Aa = a - a(0). First, take the absorption coefficient to be constant along length L of the absorbing material. This is plausible for thin materials, where diffusion equalizes photocarrier density. Let us also assume that the absorption is large enough so that multiple surface reflections can be neglected. Then the transmission is

T = (1 - R,)(1 - R,)e-"L

(36)

where R, is the front surface reflectance, R, is the reflectance of the back surface, and L is the material length. In this case, the fractional transmission change and Aa have a simple relation:

Sometimes multiple surface reflections cannot be neglected. They may even be used to enhance the nonlinear optical effect. In this case, it is more difficult, to relate absorption to transmission. The material becomes a Fabry-Perot cavity filled with a nonlinear medium -a nonlinear FabryPerot (Fig. 5). If the absorption coefficient is approximately constant along

1 RESONANT OPTICAL NONLINEARITIES IN SEMICONDUCTORS 15

FIG. 5. A nonlinear Fabry-Perot.

the length of the Fabry-Perot, the transmission is T=

( 1 - RjX1 - Rb)e-"L 1 ( 1 - Re,,)'

1 + Fsin'6

(38)

and the reflectance is

where R e f f= e-"L(RjRb)"' is the effective mean reflectance, F = 4 R e f f / (1 -Reff)' is called the coeBcient ofJnesse, and 6 is a wavelength-dependent phase, proportional to the optical path length in the Fabry-Perot, that is,

where L,,, is the length of the cavity, and ncavis the average refractive index. Transmission and reflection spectra exhibit a series of maxima and minima where the intracavity phase is a multiple of n/2. Equations (38) and (39) can be inverted to give the absorption coefficient and refractive index. For example, Eq. (38) gives a=

1 b - Jb' -L

(

- 4R, RbT2 2RjRbT

)

(41)

where L is the length of the absorbing material, and b is given by b = (1 - RjX1 - Rb) 4-2 J R / R , T [ 1 - 2 Sin2(6)]

(42)

If n is known, a can be found from transmission or reflectance measurements. If n is unknown, having 6 is good enough, and it may be possible to determine 6 from the positions of minima or maxima in the transmission or

16

ALAN KOST

reflection spectrum. Similarly, n can be found from transmission or reflection if a is known. In general, it can be difficult to accurately determine a or n near the absorption edge of a photoexcited semiconductor, where both a and n are changing. Note that there are different conventions for choosing the value of L to be used to calculate the absorption coefficient for multiple quantum well materials. Some authors set t equal to the total length of the wells without the barriers. Others use the length of the wells plus barriers. Wells and barriers are about the same thickness, so the calculation with wells plus barriers gives a result that is smaller by about a factor of 2.

IV. Resonant Optical Nonlinearity in GaAs Quantum Wells Band-filling optical nonlinearities have been more thoroughly studied in GaAs that in any other semiconductor. Band-filling nonlinearities in GaAs are also of technological importance because they are compatible with the use of diode lasers. This section summarizes studies of GaAs multiple quantum well structures (Kawase et al., 1994; Lee et al., 1988) in order to illustrate the measurement and characterization of band-filling optical nonlinearities. Quantum wells have a very large excitonic component to their absorption, which gives large optical nonlinearity at low levels of optical excitation. Both density-dependent and intensity-dependent band-filling nonlinearities are discussed in this section. In principle, having either the density or intensity dependence, the other can be found with Eq. (19) or (21). In practice, uncertainty in the values for carrier lifetime makes this difficult. If an application uses short optical pulses (much shorter than the recombination time), it is best to know the density dependence. Density-dependent absorption, with the Kramers-Kronig relation, also gives the density-dependent refractive index. if an application uses CW excitation or pulsed excitation where the pulses are much longer than the recombination time (.quasi-CWexcitation), the intensity dependence for the nonlinearity is most helpful.

1.

SAMPLE DESIGNAND FABRICATION

The cross section for the five multiple quantum well (MQW) wafers used in these studies is, shown in Fig.6. Substrates were n-type, (100) oriented GaAs. The first layer grown was a thin GaAs layer to smooth the

1 RESONANT OPTICAL NONLINEARITIE~ IN SEMICONDUCTORS

17

FIG. 6. The multiple quantum well structures used for nonlinear optical studies.

substrate. This was followed by a 0.5-pm Alo.soGao.50Aslayer with high A1 content to serve as an etch stop layer for substrate removal. The MQWs consisted of enough periods of the alternating structure to yield 0.5 pm of MQW material. The GaAs wells were 50, 75, 100, 150, or 2508, wide. The A10.32Ga0.68As barrier layers on either side of the wells were 1008,thick for all but the sample with 50-8, wells. For this sample, the barrier thickness was 1508, to minimize wavefunction overlap between the wells. The MQW material was separated from the sample surface and from the etch stop layer by 0.5-pm layers of A10.3,Gao.68As.The samples were grown by atmospheric pressure MOCVD at a temperature of 700°C and nominally undoped. For pump-probe measurements, pieces of wafer with loo-, 150-, and 250-A wells were glued to glass, MQW side down, with a transparent wax. The absorbing semiconductor substrates were removed using mechanical polishing, a rapid nonselective etch, and a final selective etch (Logan and Reinhart, 1973). A number of approaches have been developed for removing GaAs substrates. Multiple etch stop layers have been used to enhance the flatness of the remaining MQW epitaxial layers (Jewel1 et al., 1983). Dry etching of GaAs has been demonstrated with a selectivity of 2001 over A1,~,Ga0~,As(Hikosaka et al., 1981). Dilute H F was used to etch into an underlying AlAs layer, allowing GaAs/AlGaAs epitaxial layers to be peeled from the substrate (Yablonovitch et al., 1987). HF etched AlAs with a selectivity of lo7 or better over A1,,Ga0~,As in this "lift-off procedure. A single SiO, layer was deposited as an antireflection coat to the exposed

18

ALAN KOST

semiconductor surfaces. The residual reflectances were estimated to be R , = 0.03-0.06 and R, = 0.16. For low-intensity absorption and single-beam nonlinear absorption measurements on all five MQWs, the same procedure was used, except that the MQW side also was AR coated (before gluing). For these samples, residual reflectances were estimated to be Rf = 0.08 and R, = 0.1.

2. LINEAR OPTICAL ABSORPTION The linear (low-intensity) absorption properties of the MQW samples were measured with a spectrometer. The room-temperature spectra are shown in Fig. 7. Quantum confinement of electrons and holes in quantum wells gives rise to an absorption spectrum that is fundamentally different from bulk semiconductors. First, the density of states for the conduction and valence bands becomes a series of steps, also called subbands. The positions of these subbands are a strong function of the width of the quantum well. Note how the absorption edge shifts to progressively shorter wavelengths as the well width decreases. Confinement also increases the binding energy and absorption strength of excitons, making them further from the absorption of the energy bands and more pronounced. Note the pronounced absorption peaks in Fig. 7 from excitons at each step in the joint density of states. At room temperature, these features are unique to MQWs. Where two absorption peaks are close together, they can be attributed to separate transitions involving light and heavy holes, denoted Ih and hh,

7

o

o

m

~

M

a

m

WAVELENGTH (nm)

FIG. 7. The low-intensity, room-temperature absorption for the five multiple quantum well samples.

1 RESONANTOPTICAL NONLINEARITIE~ IN SEMICONDUCTORS

19

respectively. The splitting of the energy for light- and heavy-hole transitions is another consequence of quantum confinement. For the wider, less confining wells, the light- and heavy-hole excitons have nearly the same energy and cannot be resolved separately. The notation n = 1, n = 2, etc. refers to transitions between the n = 1 hole subband and the n = 1 electron subband, etc. Interband transitions between subbands with different n values are symmetry forbidden. Note that the n = 2 and higher exciton peaks are less pronounced as a consequence of reduced binding energy. None of the higher peaks could be resolved into separate light- and heavy-hole components.

3. DENSITY-DEPENDENT ABSORPTIONAND REFRACTIVE INDEX a. Measurement Procedure The experimental setup is pictured in Fig. 8. Photocarriers were pumped into the materials with 1.518-eV pulses from a mode-locked dye laser (Styryl 9-M dye) amplified up to SON at a 10-Hz repetition rate. Broadband luminescence from a cell with Styryl-9 or Styryl-13 dye was used to probe the sample. The pump and probe spots were large, about 1 and 0.5 mm in diameter, respectively. The pump pulse duration was approximately 10 ps, much shorter than the photocarrier recombination time. The probing luminescence pulses were about 5ns in duration and shorter than the estimated electron-hole lifetime in all cases, except possibly for the material with 150-8, wells under high excitation. The probe arrived at the sample

PROBE

DYE LASER

I MODE-LOCKED ARGON L A S R

x NdYAO

FIG. 8. A pumpprobe system of nonlinear optical measurements. The dye amplifier was pumped with the second harmonic of a Nd:YAG laser. Unamplified pulses were attenuated with an amustooptic modulator.

20

ALAN

KOST

within 1 ns of the pump to sample the change in transmission immediately after photocarriers were created by the pump and before they could recombine. Probe transmission was measured with an optical multichannel analyzer. All measurements were at room temperature. The absorption coefficient was determined from the measured transmission using Eqs. (41) and (42) for a nonlinear Fabry-Perot, with L equal to the total thickness of the quantum wells without the barriers. Local maxima and minima in the transmission, corresponding to Fabry-Perot resonances and antiresonances, were used to determine the intracavity phase. Refractive index changes were measured directly using the pump-induced shift in Fabry-Perot resonances and antiresonances. To a good approximation, the relation between the change in the average refractive index and a shift of the mth order resonance or antiresonance is

The refractive index change was attributed to the quantum wells, and a refractive index change for the quantum wells was defined:

6. Density- Dependent Absorption Spectra Figure 9 shows the density-dependent absorption for the three quantum well materials. The dashed lines represent the best estimate for the absorption in regions where the measurements were masked by strong pump light. These estimates were used for a calculation of the nonlinear refractive index from the Kramers-Kronig relation. Time-resolved photoluminescence measurements indicated that the electron-hole lifetime for the material with 150-Awells was about the same as the probe pulse duration at the highest excitation levels. For this sample, the amount of absorption saturation may have been slightly underestimated because the absorption spectrum recovered during the probe. The spectra show that the excitonic absorption in the lower subbands saturates at lower densities than in the higher subbands. This is expected because, for lower densities, only the lower-lying states are filled. The transitions between the higher-lying states that make up the higher excitons are not blocked. Absorption saturation for the higher excitons is primarily from Coulomb screening, which affects all excitons.

1 REWNANT OPTICAL NONLINEARITIES IN SEMICONDUCTORS

21

N (4) c 5 x 1016 1) 9.4 x 1015 (2) 6.3 x 101e

P

lz;;%J (6) 0 2 x 1017 (7) 1 . 7 ~ 1018

(0) < 5 x 101s (1) 2.3 x 101'

p;ig

(2) &I x 10M

(s) os x 1017 (7) 1.7 x 1018

I"'

< 5 x 1015 1) 5.8 x 101e (2) 0.6 x 1016 (s) 20x 1017 (4) 3sx 1017 (5) 6.2 x 1017 (6) 1.6 x 1016 1.40

1.45 1.50 1.55 PHOTON ENEROY (mv)

1.60

FIG. 9. Absorption spectra for three multiple quantum well samples versus the density of photoinduced electron-hole pairs.

The absorption changes extend over 150meV ( - 6 k , q . Excitation into higher-lying subbands is more prominent for the sample with wider wells because the subbands are more closely spaced for these materials. For the sample with narrower, 100-8, quantum wells, the absorption change is predominately for the lowest, n = 1 subband. For high carrier densities in the material with 250-8, wells, the largest change is at the n = 2 subband, which has more initial absorption than the n = 1 subband. c. Refractive Index Measurements

The spectra for the optically induced change in refractive index, found from the absorption change and the Kramers-Kronig relation (Eq. 30), are shown in Fig. 10. Arrows indicate the photon energy of exciton absorption peaks. Note the narrower spread of An for narrower quantum wells. The refractive index change found directly from the shift of Fabry-Perot resonances is compared with the An determined from the Kramers-Kronig relation in Fig. 11. Because the shift of a transmission resonance could not

22

ALAN KOST

1 3

1 s

1.40 1.46 lM 1.56 PHOTON ENERQY (ow

1.w

FIG.10. Spectra for the refractive index change of three multiple quantum well samples versus the density of photoinduced electron-hole pairs. The dashed line, labeled An,, is the saturation carrier density explained in the text.

FIG.11. A comparison of the refractive index change obtained from the shift of FabryPerot resonances (open boxes) and the Kramers-Kronig relation (closed circles). The solid lines are fits.

1 RESONANT OPTICAL NONLINEARITIES IN

SEMICONDUCTORS

23

be measured accurately, the values obtained from the Kramers-Kronig relation are a better estimate of An. The measurements from fringe shift support the validity of the indirect procedure using the Kramers-Kronig relation. d. Modeling

The carrier-density-dependent absorption was fit with a single saturation term plus a term corresponding to unsaturating background (Eq. 13). The saturation density N,,, was smallest for the material with 150-8, wells. The smallest N,,, might be expected for the MQW with the widest wells because subbands for this material had the smallest density of states. It was noted that the thermally broadened subbands were closest together in the MQW with 250-8, widest wells, and it was conjectured that subband overlap increased the density of states for this sample. Figure 12 shows the density-dependent absorption at the n = 1 exciton resonance for the three MQWs and a best fit with Eq. (13). The expression produced good fits for all three MQWs at all wavelengths.

-

-1.6 1016

1017

101'

1010

N (em*)

FIG. 12. Carrier-density-dependent absorption at the n = 1 excitonic absorption resonance. The solid lines are fits.

24

ALAN K o s ~

1

t

-~

1.40

1.46

1.60

1.60

1.55

PHOTON ENERGY (ow

FIG. 13. Saturation densities for the multiple quantum well samp .s. The solii lines indicate the “average” values used to fit the refractive index change.

The wavelength dependence of the saturation density (Fig. 13) was very weak, decreasing only slightly on excitonic resonance. For the MQWs with 250-A wells, N,,, increases slowly but steadily with increasing photon energy. Using “average” values for N,,,, indicated by the straight lines in Fig. 13, Eqs. (17) and (32) were used to model the refractive index. The dashed lines in Fig. 10 are the values ns, the maximum possible refractive index change, obtained from a,(E) the saturating component of the absorption. Modeling the refractive index gave the solid lines in Fig. 11. Table I summarizes the carrier-density-dependent absorption and refracTABLE I

REFRACTIVE INDEX FOR GaAs MULTIPLEQUANTUM WELLSTRUCTURES

DENSITY-DEPENDENT ABsoRPTlON AND

Well thickness (A) z, (pm-’)

~ ( p m - ’ ) N,, ( 1 0 1 7 ~ ~ - 3 ) Average N,,,(10” m-)) n,-rnax

100 1.7pm

1 50

250

0.1 4.7

1.1 pm -0.1 1.9

0.8 pm -0.2 3.7

7.8 0. I2

3.2 - 0.07

- 0.08

6.6

1

RESONANTOPTICAL NONLINEARITIES IN SFMICONDUCTORS

25

tion for the GaAs MQW materials. a, and a,,, are the saturating and unsaturating components to absorption at the n = 1 excitonic resonances. The interpretation of negative values of suns was that there would be optical gain with high excitation. The quantity n,-max is the largest value of n,(E). 4. INTENSITY-DEPENDENT ABSORPTION a. Measurement Procedure Nonlinear absorption for all five MQW materials was measured at room temperature with a CW dye laser (Styryl-9 dye). The output of the dye laser was attenuated by the use of an acoustooptic modulator. The first-order diffracted beam was selected with an aperture, giving the system provided a 2000:l dynamic range. The modulator output was pulsed for 2-ps duration for high-level excitation or for 200-ps duration for low-level excitation. In both cases, the pulse length was much longer than carrier recombination time. The repetition rate was 100 Hz.The incident and transmitted beams were measured with calibrated photodiodes. The beam was focused on the sample to a spot with a l/e diameter of approximately 10pm. An average spot intensity was calculated by dividing the power by the product of the l/e “diameters” in perpendicular directions. The Rayleigh range was much longer than the length of the MQW. The transmission spectra at four intensities are shown in Fig. 14(a) for the sample with 75-8, wells. The low-intensity spectrum shows transmission minima at 829 and 837 nm that correspond to the n = 1 light- and heavy-hole excitons. The minima at 822 and 855 nm, in the high-intensity spectrum, are Fabry-Perot antiresonances. As described previously the absorption coefficient was found with the measured transmission and Eqs. (41) and (42). The intensity-dependent absorption for the sample with 75-8, wells is shown in Fig. 14(b). The lightand heavy-hole exciton absorption resonances saturate at a moderate intensity, leaving a featureless spectrum that saturates at higher intensities. Note the increasing absorption below the absorption edge at moderate intensities from bandgap renormalization. The nonlinear spectra were similar for the other four MQW samples. b.

Modeling

Equation (16) was used to model the intensity-dependent absorption with two saturating components. Modeling was performed at the heavy- and light-hole excitonic resonances and at a wavelength in the absorption continuum above the excitons. Additional measurements were made at these

ALAN KOST

26

O3 0.0

t:

Ma

810

820 (L30 840 860 WAVELENQTH(nm)

880

870

m

810

820 1)30 840 MI WAVELENQTH (nm)

800

870

FIG. 14. (a) The transmission spectra for a multiple quantum well sample with 75-A wells at (i) 26, (ii), 520, (iii) 2600, and (iv) 16,500 W/cm2. Note the Fabry-Perot transmission minima at 822 and 855nm. (b) The absorption spectra corresponding to (a), after correcting for multiple surface reflections.

wavelengths for better fitting. The intensity dependence of the absorption at the n = 1 heavy-hole exciton peak is shown in Fig. 15 for the sample with 50-Awells. The solid line through the data is the model, and the lower lines are the separate saturating components. Equation (16) fits the data well at most wavelengths near the absorption edge.

c. Procedure to Determine Saturation Parameters The four parameters in Eq. (16) were determined from very low and very high intensity measurements. When I >> I",", the first term in Eq. (16) can be neglected, and Eq. (16) can be rearranged to give

P J SEMICONDUCTORS 1 REWNANT OPTICAL NONLINEARITIES

27

FIG. 15. Intensity-dependent absorption at the n = 1 heavy-hole excitonic absorption resonance for a multiple quantum well sample with 50-A wells. The solid line through the data is a fit. The lower solid lines show separate contributions to the fit.

Using the data measured at high intensities, ab,B and 1;: were found from the slope and intercept of a linear fit to l/a(l). Figure 16(a) shows the linear fit for the data at the n = 1 heavy-hole excitonic resonance for the MQW with 50-A wells. When I << I::, the second term in Eq. (16) is approximately equal to ap. Then Eq. (16) can be rearranged to give 1 --1 --exI ex + a - utg a. I,,, ag Using data measured at low intensities, atx and were found from the Figure I.16(b) shows the slope and intercept of a linear fit to l/[a(l) - @ linear fit for the data at the n = 1 heavy-hole excitonic resonance for the M Q W with 50-A wells. This procedure gave the solid lines in Fig. 15.

d. Measurement Summary Table I1 summarizes the nonlinear absorption for all five samples. Values are given for both light- and heavy-hole excitons for the samples with 50-A and 75-8, quantum wells, where the light-hole exciton could be measured separately from the heavy-hole exciton. The nonzero values for a: when measured in the absorption continuum can be interpreted as a saturation of

28

ALAN K m

i

0.0

0.5

1.o

.s

1

2.0

W T E m (WICmt)

FIG. 16. Curves illustrating the procedure for determining the fitting parameters for the intensity-dependent absorption.

B Coulombic component to the absorption. At wavelengths near the excitons, this component was probably overlapping exciton absorption tails. A t wavelengths far removed from the excitons, it was probably the Coulomb enhancement of the energy bands. Lateral carrier diffusion increased the saturation intensity somewhat. The area of the laser spot used in these measurements was approximately 100 pm2. Using the diffusion coefficient D = 17 cm2 for bulk GaAs and 20 ns as an estimate of photocarrier lifetime, an estimate for the area occupied by

29

1 RESONANT OPTICAL NONLINEARITIES IN SEMICONDUCIORS TABLE I1

INIENSITY-DEPENDENT ABSORPTION FOR GaAs MULTIPLE QUANTUM W ~ STRUCTURES L

75 100 67 839 829.5 810 1.35 1.67 1.02 1.58 0.14 1.88 960 342.5 2560 250.4 156 34.25

50 150 100 825 815 797 1.88 1.25 0.83 2.05 0.0 2.54 321 16.1 337 16.5

53.3

150

250 100 20 870

100 100 50 854

33 862.5

831 1.35 1.20

860 0.68 1.01

830 0.47

0.32 0.72 304 136.8

0.3 1 1.10 697 55.0

0.19 0.99 250 163.0

I520 59.0

100

1035 52.7

0.49

2170 172.8

+

the photocarriers is 100 pmZ 4 n D ~= 530pm2. Thus measurements with very large spots would be expected to give much smaller saturation intensities.

V. Summary of Band-Filling Nonlinearities 1. BULKSEMICONDUCTORS AND QUANTUM WELLS Table 111 selectively summarizes band-filling optical nonlinearities for bulk semiconductors and for semiconductor quantum wells. In all cases, the nonlinearities were observed at wavelengths very close to the absorption edge. Measurements were made at room temperature unless otherwise noted. Absorption and saturation parameters are listed for simple modeling. Most of the references contained only enough information to use the simplest models (Eqs. 12 and 15). For those cases where data could be modeled with multiple saturating components, they are listed with the corresponding saturation levels. Nonlinear absorption and nonlinear refraction are assumed to have the same saturation levels. Overall, absorption coefficients for small-bandgap semiconductors are saturated with less carrier density and less optical intensity. In contrast,

TABLE 111 BAND-FILLING NONLINEARITIE~ FOR BULKSEMICONDUCTORS AND SWICONDUC~OR QUANTUM WELLS Reference ZnSe CdZnSe/ZnSe (single QW) GaAs MQW

458 508

GaAs MQW

862

Bulk GaAs InAsIGaAs (superlattice MQW) InGaAs MQW

825

872 977

1604

InGaAs

InGaAs InAs (90 K) lnSb (77 K) HgCdTe (175 K)

1675 3046 5435 10600

4.5 1 (exciton) 0.27 (background) 1.88 (exciton) 1.25 (background) 1.1 (saturating)

6.25

3.2

-0.1 (unsaturating)

co

1

.I .8^1

1.2

0.45 (exciton) 0.23 (background) 0.19 (unsaturating) 0.1 (exciton) 0.17 (background) 0.07 (unsaturating) 0.27 0.07 (unsaturating) 0.37 0.024 0.0096

200 3.4 (exciton) 23 (background) 0.32 (exciton) 16.1 (background)

3.7

460 515

-0.12 Peyghambarian et al., 1988 -0.01 1 (exciton) Houser and Garmire, 1994 -0.021 (background) Lee et al., 1988

867

-0.07

Kawase et al.. 1994

874 979

-0.07 -0.06

Lee et al., 1986a McCallum et al., 1991

0.125 (exciton) 4 (background)

Fox et al., 1987b

0.5 (exciton) 5 (background)

Fox et al., 1987b

oc,

2

co 0.2

1.6 0.125 0.05 0.18

1685 3096 5450

-0.12 -0.02

-0.1 -0.08

Fox et al., 1987a Pool and Garmire, 1985 Miller et al., 1981a Hill et al., 1982

1 RWNANT

OPTICAL

NONLINEARITIES IN SEMICONDUCTORS

31

the measured changes in refractive index are relatively independent of wavelength. These seemingly contradictory observations can be reconciled by noting that small-band-gap semiconductors have smaller absorption coefficients because of a smaller density of states. Each photocarrier blocks a transition with an absorption cross section that increases with wavelength. Integrated over the spectrum, photocarriers block more absorption in small-band-gap semiconductors, and the Kramers-Kronig relation shows that all absorption changes contribute to the change in refractive index. Both bulk semiconductors and quantum wells have been modeled with an absorption that has two components, one of which saturates at low levels and is attributed to excitons. The difference is that quantum wells have a larger excitonic component. Using semiconductor quantum wells is an advantage for applications that require moderate changes in absorption or refractive index with low-level excitation or require the largest possible changes with high excitation.

2. INTERSUBBAND ABSORPTIONSATURATION Transitions between quantum well subbands give strong absorption in the infrared (Fig. 17). Intersubband absorption in quantum wells was first reported by West and Eglash, who predicted large optical nonlinearity with resonant excitation (West and Eglash, 1985). Julien and colleagues (Julien et al., 1988) measured intersubband absorption saturation near 10pm in the conduction band of GaAs MQWs with AI,.,Ga,.,As barriers doped n-type to provide electrons for the lower subband. The saturation intensity was 340kW/cm2 at 10.45pm for an MQW with 85-8, GaAs wells. The absorption coefficient was -4000cm- l , giving a nonlinear absorption coefficient a2 of about -0.01 cm/W. The relaxation time was estimated to be 10 ps, so Nsa,was about 4 x 10” electrons per cubic centimeter. Selection rules limit the use of intersubband nonlinearities for nonlinear optical devices. Only light with an electrical field perpendicular to the plane of the quantum wells is absorbed. (For a comprehensive discussion of this and other device issues in the context of photodetectors, see Levine, 1993.)

FIG. 17. Infrared optical absorption by quantum well intersubband transitions. The optical electric field must be perpendicular to the plane of the quantum wells.

32

ALAN KOST

The finite depth of semiconductor quantum wells limits intersubband absorption to wavelengths of about 3 pm or longer. 3. QUANTUM DOTSAND SEMICONDUCTOR DOPEDGLASSES

The search for larger optical nonlinearities first led to the investigation of semiconductor quantum wells and then to lower-dimensional semiconductors- quantum wires and quantum dots. Quantum dots in particular are predicted to have large optical nonlinearity (Schmitt-Rink et al., 1987; Banyai et al., 1988) (see also Volume 59, Chap. 3). There have been few experimental studies of optical nonlinearities in the lower-dimensional semiconductors because of the difficulty in fabricating collections of uniformly sized quantum wires or dots. Semiconductor-doped glasses have been investigated for some time as commercially available, potentially inexpensive nonlinear optical materials. Glasses with crystallites less than 100 A in diameter (not commercially available) show signs of quantum confinement in all three dimensions, but the variation in crystallite size is at least 10 to 15%, so the quantum size effects are washed out. Fast surface recombination makes it difficult to measure optical nonlinearity in crystallites with diameters less than 100 A, and photodarkening can be a problem at high levels of excitation. Jain and Lind used four-wave mixing with 10-ns, visible, optical pulses to measure the third-order nonlinear optical susceptibility for commercial glass filters (Jain and Lind, 1983). They measured ,y3 5 10-8-10-9 (esu) for large ( > lWA) CdSeS crystallites. Hall and Borelli used 1 to 5-ps optical pulses to measure absorption saturation at visible wavelengths in glass doped with 2 l00-A CdSeS microcrystallites (Hall and Borelli, 1988). They report a saturation fluence of 0.7 mJ/cm2 and a linear absorption coefficient of 45 cm- *. Doped glasses have relatively small absorption coefficients because the crystallites occupy only a small fraction of the total volume. Peyghambarian and colleagues (1989) used 1004s optical pulses to demonstrate absorption saturation at 10 K in doped glasses with 26- and 38-A C'dSe crystallites.

-

3. OPTICAL MODULATORSAND ACTIVEMEDIA

The optical power handling of a semiconductor electroabsorption modulator is limited by saturating absorption in the electroabsorptive region of the device. A t low powers, photocarriers created by optical absorption have

1 RESONANTOPTICAL NONL~NEAR~T~ES IN SEMICONDUCTORS

33

little effect on the absorption, and they are swept out quickly by applied fields. At high powers, large numbers of photocarriers screen the applied fields, accumulate in the electroabsorptive region, and saturate the absorption. Figure 18 shows the results of a study of exciton saturation in semiconductor quantum well modulators with 95-A GaAs wells and Al,Gal -,As barriers (Fox et a/., 1991). Saturation intensity increased with increasing applied voltage, decreasing barrier width, and decreasing A1 composition. The results were attributed to photocarriers accumulating in the quantum wells when small electric fields were screened by photocarriers and when tall, wide AlGaAs barriers impeded photocarrier motion. A built-in electric field was shown to reduce the intensity for absorption saturation in a semiconductor n-i-pi structure (Kost et al., 1991). The effect was attributed to longer recombination times for electrons and holes that were spatially separated by the field. Band-filling nonlinearities are also seen in semiconductor amplifiers and semiconductor lasers. Hot carriers, not in thermal equilibrium with the lattice, play an important role, and optical nonlinearity is described as a nonlinear gain. Nonlinearities in active media and applications to optical switching are discussed in Volume 59, Chap. 2.

t

A

0.4

66

V. I

0

4

8

12

16

REVERSE BIAS (V)

FIG. 18. Exciton saturation intensities for GaAs/AI,Ga, -,As multiple quantum well optical modulators. The GaAs well width was 95 A in all cases. The number of periods was adjusted to make the total multiple quantum well thickness equal 1.0pm.

34

ALAN Kosr

VI. Figures of Merit Figures of merit are often used to indicate the size and usefulness of optical nonlinearities. Saturation intensity and saturation density are examples. This section describes other commonly used figures of merit. Nonlinear Absorption

u.

The nonlinear absorption coefficient a2 is defined by a(1) = a.

+ a21

(47)

or da a2 3 -

dl

These expressions best describe absorption that changes linearly with intensity when they both give the same value for a2, a constant equal to the absorption change divided by the intensity. For intensities well below saturation, these equations are equivalent to Eq. (15), with a2 given by - aO/lsat.When the absorption does not increase linearly with intensity, Eqs. (47) and (48) are used to define an intensity-dependent a2. The nonlinear absorption cross section is defined by

It is the amount of absorption change, at photon energy E, per electron-hole pair per unit volume.

h.

Nonlinear Refraction

The nonlinear refraction coefficient n2 (“n two”) is analogous to the nonlinear absorption coefficient a2 and is defined by

n2

3

dn dl

-

1 RESONANT OPTICAL NONLINEARITIES IN SEMICONDUCTORS

35

The corresponding third-order nonlinear susceptibility x3 is given by (Boyd, 1992)

4 x3[esu] = 0.0395 n2

[g]

An important consideration for devices that use resonant nonlinear refraction is the overlapping absorption and the figure of merit n2/u or x3/u. The refractive volume is the refractive index change per electron-hole pair per unit volume:

c.

Response Time

Band-fillingoptical nonlinearities turn on as soon as enough electron-hole pairs are created. This can occur in less than a hundred femtoseconds with pulsed excitation. The effects will be dynamic for a few picoseconds (or longer at low temperatures) as excitons ionize, free electron-hole pairs thermalize among themselves in the energy bands, and the thermalized populations assume the temperature of the lattice (Gong et af., 1990) (see also Chap.3). The fast dynamics can be important for nonlinear optical devices. The impact on saturable absorbers for laser modelocking is discussed in Volume 59, Chap. 4. With CW light, the turn-on time is usually on the order of the electron-hole recombination time, typically nanoseconds. The turn-on and recombination times are similar because particles must be created at least as fast as they recombine in order to achieve a substantial photopopulation. Very high CW excitation can create particles much faster, but practical considerations limit the amount of available power. Band-filling nonlinearities turn off when the excess electron-hole population decays. The decay is usually by recombination in nanoseconds. Ion implantation (Silberberg et al., 1985), surface recombination (Lee et af., 1986b), and low-temperature epitaxial growth (Siegner et al., 1996) have been used to reduce the recombination time to below a nanosecond. Electrical fields have been used to sweep photocarriers from a quantum well in 130ps (LiKamWa et al., 1991), and intervalley scattering in type I1 quantum wells has been used to remove electrons from the wells in 2ps (Feldman et a/., 1990). Response time is taken into account by defining figures of merit like n,/az. The turn-off time is most often used, because turn-on can be arbitrarily fast.

36

ALAN KOST

An examination of Eqs. (21), (50), and (53) shows that n,/a? = ne,/ho. For some applications, it is best to sacrifice speed in order to achieve high degrees of parallelism.

d. Experimental Values and Trends Table IV gives figures of merit for band-filling optical nonlinearity with various bulk and quantum well semiconductors. The values for T are approximate electron-hole recombination times at low levels of excitation. They are a conservative estimate of turn-off time. Overall, the magnitude of the figures of merit increases with wavelength. Small values for the nonlinear cross section and refractive volume for the CdSnSe/ZnSe single Q W are probably from an overestimate of the recombination time. There are not enough data in Table IV to see a trend for the nonlinear cross section, but it should increaserapidly with wavelength.Transitions between states in valence and conduction bands have an absorption strength proportional to ?,P:c,where 1is the transition wavelength,and P$ is the square of the interband momentum matrix element. Pv’, is approximatelyconstant for most semiconduci tors (Yu and Cardona, 1996). The probability that any given electron or hole will block a transition at a specified wavelength is proportional to l/w3”[see Eq. (3) and related discussion], and m,increases approximately like l/A. Therefore, the magnitude of the nonlinear absorption cross section should increase like

The refractive volume nch also should increase with wavelength, but somewhat less rapidly because the distribution of electrons and holes in the bands does not have the same effect. All blocked transitions contribute to the nonlinear refractive volume at any given photon energy. In Table IV, the magnitude of the nonlinear refractive volume nch increases from 0.4 x 10-*9cm3at 1 = 0.87pm (GaAs) to 8 x 10-19cm3at 2 = 3.1 pm (InAs). The data in Table IV show no trend for a2. but the magnitude of the nonlinear refractive coefficient n2 increases by four orders of magnitude as the wavelength increases from 0.45 to 10.6pm. It is difficult to predict the wavelength dependence for a2 and n2 because they depend on photocarrier lifetime. The recombination time T from Table IV increases rapidly with decreasing bandgap, so a2 and n2 also might be expected to increase, but the values for T represent with low-level excitation. Auger recombination reduces carrier lifetime at high photocarrier densities, especially for small-bandgap semiconductors (Agrawal and Dutta, 1993).

TABLE IV FIGURES OF MERIT FOR BAND-FILLING OPTICAL NONLINEARITY IN BULKSEMICONDUCTORS AND SEMICONDUCTOR QUANTUM WELLS

ZnSe CdZnSefZnSe (single QW) GaAs MQW

0.46 0.5 1

-0.1

0.83

-29 (exciton) -0.4 (background)

GaAs MQW Bulk GaAs InAsfGaAs (superlattice MQW) InGaAs MQW InGaAs InGaAs InAs (90 K) InSb (77 K) HgCdTe ( 175 K)

0.86 0.87 0.98 1.6 1.7 1.7 3.1 5.4 10.6

-0.3

-60

-4 -0.1

-2.9 -0.6 -4.3

-3.8 x 10-7

-2.1 x 10-6

- 1.2 -0.03

- 1.0 -0.4 -2.5

0.03 0.2 20

Lee et al., 1988

10 8

Kawase et al., 1994 Lee et al., 1986a McCallum et al., 1991

-3 x 10-4

-4

-240

-4 x 10-5 -1 x 10-4 -1 x 10-3 - 7 x 10-3

-8

Peyghambarian et al., 1988 Houser and Garmire, 1994

4.5 220

400 lo00

Fox et al., 1987b Fox et al., 1987b Fox et a/., 1987a Pool and Garmire, 1985 Miller et al., 1981a Hill et al., 1982

38

ALAN Kosr

VII. Optical Nonlinearity from Free Carrier Absorption and Refraction

BASICEQUATIONS

I.

Free carrier absorption refers to optical transitions between states in the same band (Fig. 19). These transitions do not conserve wavevector and must involve interactions with phonons, crystal imperfections, or other change carriers. To model free carrier absorption and refraction, we begin with expressions for the real and imaginary parts of the complex refractive index n + ik:

and k=

{&

[(E:

+ Ef,”’

11’

- &J}

where E~ and c2 are the real and imaginary parts of the dielectric constant, respectively, and k is related to the absorption coefficient by

The classic Drude model for the dielectric constant gives (Wooten, 1972)

-=“;-(,) El

aoc

--Ne’ mcffEO

EO

T2

[(or)’ + 11

t

WAVEVECTOR

Fib. 19. Free cwrier ahsorprion refers to an optical transition within an energy band. The transitions must include an interaction with phonons, crystal imperfections, or other charge cilrners to conserve wavevector.

1 RESONANTOPTICAL NONLINEARITIES IN SEMICONDUCTORS

39

and

Ne2

nc _c2 ---ao+60

T

meffco“ T ) 2

0

+ 11

(59)

where N is the density of electrons or holes, me, is the effective mass of an electron or hole, o is the optical frequency, and r is the mean time between scattering events for a particle. no and a. are the refractive index and absorption from all other effects. At optical frequencies, the optical field oscillates many times between carrier collisions, so (or)’>> 1. Free carrier contributions to are usually small, so n !z no. Also, it is usually the case that uA << 4nn. Solving Eqs. (55) to (59) with these approximations gives the following expressions for free carrier absqption and refraction: afc((n)=

R2Ne3 4n2c3nocom~ffp

and n“(A)

=

-

A2Ne2 8n2c2noeomeff

where the scattering time r has been replaced with the carrier mobility R2 dependence for both quantities suggests that these free carrier effects are more important at longer wavelengths. Equations (60) and (61) give quantitative values that may differ from experimental values by more than an order of magnitude in the near- to midinfrared (A !z 1 to 10pm). The wavelength dependence is also not quite correct. Empirically, free carrier absorption is observed to increase as A”, where p can range from 1.5 to 3.5. The deviation from the A2 dependence can be understood as arising from a frequency-dependent carrier scattering time T, with the value of p depending on the scattering mechanisms (Seeger, 1991). Experimental values for free carrier absorption are listed in Table V. p = eT/m,,,. The

2. NONLINEAR OPTICAL SUSCEPTIBILITIES Optical nonlinearities arise from a light-induced modification of one or more of the parameters on the right-hand side of Eqs. (60) and (61). For example, light-induced free carrier absorption and refraction occur if inter-

40

ALAN KOST TABLE V FREE CARRIER ABSORPTO IN

AT

9p

l FOR

TYPE SEMICONDUCTORSa

Carrier Concentration Material

(1017 ~

m 3)-

arC/N(iO”m-*) ~

GaAs InAs

GaSb lnSb InP GaP AlSb Ge

1-5 0.3-8 0.5

1-3 0.4-4 10 0.4-4 0.5-5

3 4.7 6 2.3 4 32 15

-4

P ~~

3 3 3.5 2 2.5 1.8 2 -2

“From Fan. 1967.

band absorption increases the free carrier density N . Alternatively, optical excitation or heating that redistributes carriers within a band will modify the effective mass in semiconductors with nonparabolic bands. Optically induced transitions of holes between valence bands also can modify the effective mass. Optical nonlinearity from these effects (summarized in

FIG. 20. Sources of optical nonlinearity from free carrier absorption or free carrier refraction. (a) The creation of electron-holepairs by excitation across the bandgap. (b) Optical o r thermal excitation or of particles within a band. (c) intervalence band transitions.

1

RESONANT OPTICAL NONLINEARITIES IN SEMICONDUCTORS 41

Fig. 20) is most important at long wavelengths, where free carrier absorption and refraction are large. Auyang and Wolff have measured free carrier nonlinear optical susceptibilities at 10.6 pm for a variety of semiconductors using four-wave mixing (Auyang and Wolff, 1989). Table VI lists three of their measurements for which both free carrier absorption and refraction gave optical nonlinearity and a band-filling measurement chosen for comparison (Miller et al., 1984). The largest x3 is for band filling, but the recovery time was much longer. Taking response time and background absorption into account, all these free carrier nonlinearities have about the same figure of merit ~ ' l a . 5 .

3. OPTICAL SWITCHING OF MICROWAVES Free carrier absorption and refraction also have been used to demonstrate an optically controlled microwave switch (Kost et al., 1993). The concept is illustrated in Fig.21. Light is used to generate electron-hole pairs in an n-i-p-i material. A free-space (unguided) microwave pulse is attenuated by the charges, which interact strongly with microwaves. An n-i-p-i material is used because it is very sensitive to optical illumination. In an n-i-p-i material,

n-I-p

MICROWAVE INPUT

llG

FIG. 21. An optically controlled microwave shutter.

TABLE VI RE.WNANT FREE CARRIER OPTICAL NONLINEARITIES FOR SMALL-BAND-GAP SEMICONDUCTORS AT 10.6 /All’

Material n-lnSb P

N

P-Hb,-,Cdo.2,Te HgTe Hg,.,,Cdo.,,Te “From Auyang and Wolff, 1989.

Nonlinear Mechanism Conduction band nonparabolicit y lntervalence band transitions lnterband transitions heating Band filling

x3/15

+

x’ (esu)

i (pm)

T(K)

10.6

2

10.6

300

8 x lo-”

10.6

300

2x

10.6

80

2

to-’

6

ci (cm-’)

T

(sec)

(esu . m/s)

4x

3 x to4

48

2 x 10-1’

8x

3400

5 x lo-”

9 x 103

24

3 x 10-6

8 x 104

2

lo5

1 RESONANT OPTICAL NONLINEARITIE~ IN SEMICONDUCTORS

43

photogenerated charge is spatially separated, electrons into n-layers and holes into p-layers, so recombination times are long (microseconds to milliseconds). An illumination of less than 1 W/cm2 can produce 10l8cm-3 electron-hole pairs. For the demonstration, a GaAs n-i-p-i structure, grown on a semiinsulating GaAs substrate, was illuminated with the CW output of an argon-ion laser. An optical intensity of 800mW/cm2 produced a 50% change in the transmission of the microwave pulse, which contained frequency components between 10 and 80 GHz. The transmission change was attributed to a combination of absorption and reflection. Figure 22(a) shows the transmitted microwave signal with and without the laser. Figure 22(b) shows the corresponding frequency components and the transmission, which was found to be independent of frequency.

0

20

40 60 80 TIME (PICOSECONDS)

100

7

FRECUENCY (QW

0 0

20

40

80

80

100

. FREQUENCY (GHz)

FIG. 22. (a) The transmitted microwave pulse, with and without illumination of the n-i-p-i material. (b) The frequency components for the pulse in (a). Transmission was obtained by dividing the frequency spectra. The dashed line is predicted by the Drude model.

44

ALAN KOST

Although inadequate for the near- and midinfrared regions, the Drude model accurately describes free carrier absorption and refraction in the far-infrared region ( A > 50 pm) if the carrier densities are at least moderate (> 10” (Perkowitz, 1971). Modeling of the photoinduced microwave attenuation in the n-i-p-i material with Eqs. (54) to (58) but without the approximation (OH)’ >> 1 gave the dashed curve in the inset of Fig. 22(b).

VIII. Optothermal Optical Nonlinearities The band gap of most semiconductors decreases with increasing temperature at a rate of -0.5 meV/K. As a result, material heating can be a problem for devices that use resonant semiconductor optical nonlinearities. The shift of the absorption edge can move operating wavelengths away from the wavelength of the laser. Considerable effort has been made to remove heat from devices in order to minimize the effects of heating (Massebouef et al., 1989). On the positive side, optothermal effects can be used as the basis for semiconductor optical nonlinearities. Optically induced changes in refractive index have received the most attention. Near the band edge, the temperature dependence of the refractive index can be written as the sum of two contributions (Wherrett et al., 1988):

dn a E , ---+($) d T aE, d T

dn --

nr

The first term represents the contribution from shifting absorption edge. It dominates near the absorption edge. The refractive index change from this term is calculated from the thermally induced change in the absorption near the absorption edge with the Kramers-Kronig relation (Eq. 30). The second term represents all other contributions that are not resonant at the band edge. The refractive index change is usually positive just below the absorption edge and on the order of KOptothermal nonlinearities are comparable with band filling ( n 2 10 cm2/W), but they are slower. The optothermal nonlinearity turns on when sufficient heat has been deposited. The material recovers when heat diffuses from the excited volume. Turn-on and turn-off times depend on optical power, optical spot size, and the way the material is heat sunk, but typical values are from 0.1 to 10.0ms for both (Janossy et al., 1986). A number of nonlinear logic elements and bistable devices have been demonstrated based on the ogtothermal effect in ZnSe, which has a large

’.

-

1

RESONANTOPTICAL NONLINEARITIES IN SEMICONDUCTORS 45

thermal refractive coefficient (dn/dT = 2 x K-I ) near its absorption edge at 470nm (Walker et al., 1990). The devices are fabricated by evaporating alternating layers of ZnSe and a low-index dielectric to form a nonlinear interference filter.

IX. All-Optical Switching 1. THEOPTICAL COMPUTER The 1980s saw an intensive effort to develop the components for an optical computer. It was conjectured that computation speed could be increased by combining the massive parallelism of optics with the optical analogs of electronic memory and logic. There was much effort to develop the logic and memory devices using nonlinear optical effects (Gibbs, 1985; also see the articles in IEEE J. Quantum Electron., Vol. QE-21, No. 9, 1985, Special Issue on Optical Bistability) and with resonant semiconductor optical nonlinearities in particular. For an interesting overview of the state of the field in 1986, see the article by Bell (1986). The quest for a competitive optical version of digital electronic computers has not been successful. However, the heir to the optical computer may be the all-optical communications network, where the network is the “computer.” In this context, optical switching is called photonic switching (Midwinter, 1993; Hinton, 1993).

2. THENONLINEAR FABRY-PEROT If an optically nonlinear material is in a Fabry-Perot cavity, reflections give optical feedback that can lead to rapid transmission switching and hysteresis. Switching like that shown in Fig.23(a,b) is used for optical memory. The transmission abruptly increases and decreases at different levels. Between the switching levels, there are two possible transmission states. In this range, the transmission is low or high depending on the initial input. For optical memory, the two output states are taken to represent the logical values 0 and 1. An input beam, which carries this information, has intensity in the bistable range. A control beam is used to momentarily raise the total intensity and switch the transmission state high. The signal beam is momentarily removed to reset the transmission. A nonlinear Fabry-Perot can have a single valued response like the one shown in Fig. 23(c). A device with this response is an optical implementation

46

ALAN Kosr

--L INPUT -C

FIG. 23. Four types of optical switching.

of an AND gate. If two input beams are incident on the device, each just below the switching level, the output is high only if both beams are present. A device with a response like Fig. 23(d) can be used for a NOR gate, using two input beams to control the transmission of a third signal beam. The device geometries are illustrated in Fig. 24. To understand the operation of the nonlinear Fabry-Perot, note that the intracavity phase and absorption are directly related to the intracavity intensity I,,,. The relation between I,,, and incident intensity Iinc is

FIG.24. Geometries for implementing (a) optical memory, (b) A N D gates, and (c) NOR gates.

1 RESONANT OPTICAL NONLINEARITIES IN SEMICONDUCTORS 47

complex. For a lossless Fabry-Perot, the transmitted intensity is equal to intensity in the forward-going cavity wave reduced by reflection at the back mirror:

If the cavity is absorbing, it is common to define an average cavity in tensity: (Icav)

:l

-

[lcav-forward(X)

+ Icav-back(XI1 dx

(65)

The relation between (Icav)and linc is

where the absorption coefficient a is assumed to be constant along the length of the Fabry-Perot. As mentioned previously, the constant-a approximation is most plausible for thin materials, where carrier diffusion equalizes photocarrier density. Optical feedback comes from the interplay between the optical nonlinearity and Eqs. (66), (40), and (38). Consider a Fabry-Perot with a nonlinear refractive index. As the incident intensity increases, so does the intracavity intensity. The refractive index and the cavity phase change. Eventually, the phase change brings the cavity close to a transmission resonance, when there is a rapid increase in the intracavity intensity, a large change in the refractive index, and so on. If the cavity was initially far from resonance, the feedback is especially strong, transmission is locked in the high state, and bistability occurs. There are numerous publications devoted to optimizing nonlinear Fabry-Perots (Miller, 1981; Wherrett, 1984; Garmire, 1989).

3. DEMONSTRATIONS OF OPTICAL BISTABILITY AND OPTICAL LOGIC

Table VII lists demonstrations of optically bistability and optical logic operations with Fabry-Perots and resonant semiconductor optical nonlinearities. Unless otherwise noted, the devices were operated at room

TABLE VII OPTICAL SWITCHING DEVICES

Material

P

00

Device

Wavelength (pm)

Switching Intensity/ Fluence

ZnSe AlGaAs

Bistable Bistable

0.48 0.83

2.1 kW/cm’ 3 k W/cmZ

GaAs/AlGaAs MQW GaAs GaAs

NOR gate NOR gate, AND gate NOR gate

0.87 0.88 0.89

0. I p.J/pm2

InGaAsP InGaAs/InP MQW

Bistable Bistable

1.31

3.3 kW/cm2 0.1 pJ/cmz

lnAs (77 K) InSb (77 K ) HgCdTe (77 K)

Bistable Bistable X-OR gate

1.55

3.1 5.5 10.6

0. I pJ/pmz

40 W/cm2 50 W/cm2 1 kW/cmz

Comments Optothermal nonlinearity Thermally stable operation I-ps turn-on 9 x 9 arrays Surface recombination gives 30-ps recovery Semiconductor amplifier Optical communications and eye-safe wavelength

Reference Janossy et a/., 1985 Masseboeuf ef al., 1989 Migus ef al., 1985 Venkatesan et al., 1986 Lee et al.. 1986b Sharfin and Dagenais. 1985 Tai er a/., 1987 Poole and Garmire, 1985 Miller, 1981b Miller et al., 1984

1 RESONANTOPTICAL NONLlNEARlTlES IN SEMICONDUCTORS

49

temperature. When switch-up and switch-down intensities differed significantly for a bistable device, the higher of the two is listed. Switching intensities for all room-temperature devices was about 2 to 3 kW/cm2. With pulsed excitation, devices were switched with about 0.1 pJ/cm2. Switching in 1 ps was demonstrated, but the device could not switch again until photocarriers recombined in a few nanoseconds. The contrast ratio between states of high and low transmission was about 2:l to 5 1 for most devices. The InGaAsP amplifier had optical gain.

X. Summary and Conclusions Resonant optical nonlinearities in semiconductors occur at small intensity and small fluence, but the spectral overlap of the absorption and refractive index changes with the absorption edge must be taken into account for device design. There are many underlying mechanisms for the optical nonlinearities, but the material response can be described approximately with simple expressions. Figures of merit for band-filling nonlinearities are generally larger for small-bandgap semiconductors. Semiconductor quantum wells have very large optical nonlinearity at low levels of excitation and give larger absolute changes in absorption and refractive index. Optical nonlinearity from free carrier absorption or free carrier refraction is important at wavelengths of 10pm or longer. These effects have been used to demonstrate an optically controlled microwave shutter. Large optotherma1 nonlinearities occur at all wavelengths, unless efforts are made to minimize sample heating. The quest to build an all-optical computer stimulated the study of resonant semiconductor optical nonlinearities and led to the demonstration of all-optical memory and all-optical logic. In the future, resonant semiconductor optical nonlinearities may be used widely in optical communications networks.

ACKNOWLEDGMENTS I acknowledge my collaborations in this field with students, faculty, and visitors at the University of Southern California.

LISTOF ABBREVIATIONS AND ACRONYMS

cw MOCVD MQW QW

continuous wave metalloorganic chemical vapor deposition multiple quantum well quantum well

50

ALAN KOST

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Midwinter, J. E. (1993). Photonics in Switching, Vols. 1 and 11 (3. E. Midwinter, ed.). Academic Press. Migus, A., Antonetti, A., H u h , D., Mysyrowicz, A.,Gibbs, H. M. Peyghambarian,N., and Jewell, J. L. (1985). ”One-Picosecond Optical NOR Gate at Ro?m Temperature with a GaAsAlGaAs Multiple-Quantum-WellNonlinear Fabry-Perot Etalon,” Appl. Phys. Lett. 46.70. Miller, A,, Miller, D. A. B., and Smith, S. D. (1981). “Dynamic Non-Linear Optical Processes in Semiconductors,”Aduunces in Physics 30,697. Miller, A,, Parry, G., Daley, R. (1984). ”Low-Power Nonlinear Fabry-Perot Reflection in CdHgTe at IOpm,” IEEE J. Quantum Electron. QE20,710. Miller. D. A. B. (1981). “Refractive Fabry-Perot Bistability with Linear Absorption: Theory of Operation and Cavity Optimization,” IEEE J. Quantum Elecfron. QE17, 306. Miller, D. A. B. (1982). “Saturation of Band-Tail Optical Absorption in InSb,” Proc. R. Soc. Lund. A 379, 9 1. Miller, D. A. B., Seaton, C. T., Prise, M. E., and Smith, S. D. (1981a). “Band-Gap-Resonant Nonlinear Refraction in 111-V Semiconductors,”Phys. Rev. Lett. 47, 197. Miller, D. A. B., Smith, S. D., and Johnson, A. (1979). “Optical Bistability and Signal Amplification in a Semiconductor Crystal: Applications of New Low-Power Nonlinear Effects in InSb,” Appl. Phys. Lett. 35, 658. Miller, D. A. B., Smith, S. D., and Seaton, C. T. (1981b). “Optical Bistability in Semiconductors,” IEEE J. Quantum Electron. QE-17. 312. Moss, T. S. (1954). Proc. Phys. SIX. (London) B 67, 775. Olbright, G. R., Peyghambarian, N., Koch, S. W., and Banyai, L. (1987). “Optical Nonlineanties of Glasses Doped with Semiconductor Microcrystallites,”Opt. Lett. 12, 413. Pereira, M. F., Jr. (1995). “Analytical Solutions for the Optical Absorption of Semiconductor Superlattices,” Phys. Rev. B 52, 1978. Perkowitz, S. (1971). ”Far Infrared Free-Carrier Absorption in n-Type Gallium Arsenide,” J. Phys. Chem. Solids 32,2267. Peyghambarian, N., Fluegel, B., H u h , D., Migus, A,, Joffre, M., Antonetti, A., Koch, S. W., and Lindberg. M. (1989). “Femtosecond Optical Nonlinearities of CdSe Quantum Dots,” IEEE J. Quantum Electron. 25, 2516. Peyghambarian, N.. Park, S. H., Koch, S. W., Jeffrey, A,, Potts, J. E., and Cheng, H. (1988). “Room-Temperature Excitonic Optical Nonlinearities of Molecular Beam Epitaxially Grown ZnSe Thin Films,“ Appl. Phys. Let?. 52, 182. Pwle, C. D. and Garmire, E. (1985). “Bandgap Resonant Optical Nonlinearities in InAs and Their Use in Optical Bistability,” IEEE J . Quantum Electron. QE21, 1370. Raj, R., Sfez, B. G . , Pellat, D., and Oudar, J.-L. (1992). “Saturation of Band-Tail Absorption in Multiple Quantum Wells and Its Link with Nonlinear Index Saturation,” Phys. Sfat. Sol. (h) 173, 473. Schmitt-Rink,S., Chemla, D. S., and Miller, D. A. 8. (1985). ‘Theory of Transient Excitonic Optical Nonlinearities in Semiconductor Quantum-Well Structures,” Phys. Rev. B 32, m1.

Schmitt-Rink, S., Miller, D. A. B., and Chemla, D. S. (1987). “Theory of the Linear and Nonlinear Optical Properties of Semiconductor Microcrystallites,”Phys. Rev. B 35.81 13. Seeger, K. (1991). Semiconductor Physics, 5th ed. Springer-Verlag,pp. 347-360. Sharfin, W. F. and Dagenais, M. (1985). “Room Temperature Optical Bistability in InGaAsPl InP Amplifiers and Implications for Passive Devices,” Appl. Phys. Lett. 46, 819. Siegner. U., Fluck, R., Zhang, G., and Keller, U. (1996). “Ultrafast High-Intensity Nonlinear Absorption Dynamics in Low-Temperature Grown Gallium Arsenide,” Appl. Phys. Lett. 69, 2566.

1 RESONANT OPTICAL NONLINEARITIES IN SEMICONDUCTORS 53 Silberberg Y., Smith, P. W., Miller, D. A. B., Tell, B., Gossard, A. C., and Wiegmann, W. (1985). "Fast Nonlinear Optical Response from Proton-Bombarded Multiple Quantum Well Structures," Appl. Phys. k t t . 46, 701. Tai, K.,Jewell, J. L.. Tsang, W. T., Temkin, H., Panish, M., and Twu, Y. (1987). "1.55pm Optical Logic Etalon with Picojoule Switching Energy Made of InGaAs/InP Multiple Quantum Wells," Appl. Phys. Lett. 50, 795. Urbach, F. (1953). T h e Long-Wavelength Edge of Photographic Sensitivity and of the Electronic Absorption of Solids," Phys. Rev. 92, 1324. Venkatesan, T., Wilkens, B., Lee, Y. H., Warren, M., Olbright, G., Gibbs, H. M., Peyghambarian, N., Smith, J. S.,and Yariv, A. (1986). "Fabrication of Arrays of GaAs Optical Bistable Devices," Appl. Phys. Lett. 48, 145. Walker, A. C.,Wherrett, B. S., and Smith, S. D. (1990). "First Implementations of Digital Optical Computing Circuits Using Nonlinear Devices." Nonlinear Photonics (H. M. Gibbs, G. Khitrova, and N. Peyghambarian (eds.). Springer-Verlag, Ch. 4. Weisbuch, C. and Vinter, B. (1991).Quantum Semiconductor Structures. Academic Press. West, L. C. and Eglash, S.J. (1985)."First Observation of an Extremely Large-Dipole Infrared Transition within the Conduction Band of a GaAs Quantum Well," Appl. Phys. Lett. 46, 1156. Wherrett, B. S. (1984). "Fabry-Perot Bistable Cavity Optimization on Reflection," IEEE J. Quantum Electron. QE-20,646. Wherrett, B. S., Walker, A. C., and Tooley, F. A. P. (1988). "Nonlinear Refraction for CW Optical Bi-Stability." Optical Nonlinearities and Instabilities in Semiconductors (H. Haug, ed.). Academic Press, Ch. 10. Wooten, F. (1972).Optical Properties of Solids. Academic Press, Ch. 3. Yablonovitch, E., Gmitter, T., Harbison, J. P., and Bhat, R. (1987). "Extreme Selectivity in the Lift-off of Epitaxial GaAs Films," Appl. Phys. Lett. 51,2222. Yariv, A. (1989). Quantum Electronics, 3rd ed., John Wiley & Sons, pp. 232-241. Yu, P. Y. and Cardona, M. (1996). Fundamentals of Semiconductors. Springer-Verlag, p. 66.

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SEMICONDUCTORS A N D SEMIMETALS VOL . 58

CHAPTER 2

Optical Nonlinearities in Semiconductors Enhanced by Carrier Transport Elsa Garmire THAYER SCHOOLOF ENGINEERING DARTMOUTH COLLEGE

.

HANOVER NEWHAMPSHIRE

LISTOF ACRONYMS . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. INTRODUCTION ........................... 1. Local Nonlinearities Enhanced by Carrier Transport . . . . . . . . . 2. Nonlocal Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . 3. Figures of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . RESULTS ON OFI-ICAL N O N L I N E AINFLUENCED RI~ BY 11. EXPERIMENTAL

. . . . . . .

CARRIER TRANSPORT .......................... 1. Enhanced Nonlinearities Based on State Filling with Decreased Carrier Recombination Rates . . . . . . . . . . . . . . . . . . . . . . . . . 2. Enhanced Nonlinearities Based on Photomodulation of Internal Field . . . . 3. Combined Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . 4 . Self-Modulation of External Fields . . . . . . . . . . . . . . . . . . . . . . . . 111. FIELDDEPENDENCE OF THE OPTICAL PROPERTIES OF SEMICONDUCTORS 1. Absorption Spectra in Direct- Band Semiconductors . . . . . . . . . . . . 2. Franz-Keldysh Effect . . . . . . . . . . . . . . . . . . . . . . . . . 3. Kramers-Kronig Relation: Electrorefraction . . . . . . . . . . . . . . . 4. Quantum Confined Stark Effect . . . . . . . . . . . . . . . . . . . . 5 . Advanced Quantum Confined Stark Concepts . . . . . . . . . . . . . . . 6. QCSE at Other Wavelengths . . . . . . . . . . . . . . . . . . . . . 7 . Electrically Controlled State Filling . . . . . . . . . . . . . . . . . . . IV . EXPERIMENTAL CONFIGURATIONS ..................... 1. Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Absorption-Only Interferometer . . . . . . . . . . . . . . . . . . . . 3. Interferometer Based on Phase Shgt . . . . . . . . . . . . . . . . . . 4. Fabry- Perot Geometries . . . . . . . . . . . . . . . . . . . . . . . 5 . Bragg Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Four- Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . . V . CHARACTERISTICS OF EXPERIMENTAL DEVICES THAT UTILIZE SELF-MODULATION . 1. Speed of nipi Structures . . . . . . . . . . . . . . . . . . . . . . . . 2. Modeling Type nipi Structures . . . . . . . . . . . . . . . . . . . . . 3. Picosecond Excitation of nipi’s . . . . . . . . . . . . . . . . . . . . . 4 . Lateral Enhanced DiJusion . . . . . . . . . . . . . . . . . . . . . . 5 . Experimental Performanceof nipi’s Inserted into Devices . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56 56 51

60 61 62 63

64 74 80 81 86 86 88 90 91 102 113 122 123 123 126 127 128 139 146 146 147 149 154 155 156 160

55 Copyright {CI 1999 by Academic Press All rights 01 reproduction in any form reserved. ISBN 0-I?-752167-4 ISSN n080-~784/ws3n.00

56

ELSAGARMIRE

LIST OF ACRONYMS CBE CQW CR

cw

FK FP IMFP KK MBE MOCVD QCSE QW SEED SLM WSL YAG

chemical beam epitaxy coupled quantum well contrast ratio continuous wave Franz- Keldysh Fabry-Perot impedance-matched Fabry-Perot K ramers-Kronig molecular beam epitaxy metallo-organic chemical vapor deposition quantum confined Stark effect quantum well self electro-optic effect device spatial light modulator Wannier Stark localization yttrium aluminum garnet

I. Introduction This chapter is based on the photoexcitation of free carriers, in which some fraction of the light incident on a semiconductor is absorbed, creating electron-hole pairs. This photogeneration of carriers causes a change in the optical properties of the semiconductor so that subsequent light sees a change in absorption and/or refractive index as a result of the existence of the photocarriers. A brief review of such semiconductor nonlinearities was published in Physics Today (Garmire, 1994). This chapter includes only resonant processes and excludes virtual excitations, in which no physical electronic transitions take place and which may be nonresonant. The requirement of real physical excitation of photocarriers means that the nonlinearities must have the speed of electronic excitation times, typically picoseconds. The nonlinearities will last as long as the optically excited carriers remain. Depending on the device structure, this can be anything from 10 ps to milliseconds. The ability to control the response time by device design is one advantage of semiconductor nonlinearities that use carrier transport, the focus of this chapter. The applications that motivate the study of these nonlinearities are those that require a high degree of parallelism, with low operating intensities and moderate speeds. That is, this study will exclude the femtosecond and picosecond virtual processes that require very high optical field strengths, and consequently iace issues of multiphoton absorption. The high degree of parallelism means that two-dimensional geometries are required, ruling out waveguides. There-

2 OPTICAL NONLINEARITIES I N SEMICONDUCTORS

1

-. -

57

Input

Z FIG. 1. Thin film epitaxial geometry for optical nonlineanties involving semiconductor quantum wells at normal incidence; two-dimensional applications include spatial light modulators. The substrate (shown as dashed lines) can be removed if necessary. Incident light is normal to the growth planes, in a direction labeled z (by convention).

fore, the geometry under consideration in this chapter is thin semiconductor films, grown epitaxially and illuminated at normal incidence, such as shown in Fig 1. By convention, the direction of crystal growth is labeled z, and the thickness of quantum wells is designated L,. Direct-band semiconductors are usually chosen, because their band edge is sharper than indirect-band semiconductors and the optical absorption between direct bands is strongly influenced by electric field. ENHANCED BY CARRIER TRANSPORT 1. LOCAL NONLINEARITIES In semiconductors, optical nonlinearities can be local or nonlocal (Garmire et al., 1989). In local nonlinearities, optical excitation changes the absorption and refractive index of the semiconductor locally; that is, photogenerated carriers stay where they are and fill available states, reducing the absorption, as discussed in the preceding chapter. The amount of state filling depends on the photocarrier density. Thus, if photocarriers can live longer, the optical intensity required to maintain the same level of optical nonlinearity can be lower. That is, the material has a higher sensitivity. Carrier recombination rates can be reduced if photogenerated holes can be separated from electrons, which reduces the probability of recombination. The localized filling of available states in bulk semiconductors (Lee et al., 1986) and in quantum wells (Chemla et al., 1984) causes a change in the absorption coefficient a that can be described heuristically by a simple

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saturation as a function of photocarrier density N (Kawase er al., 1994, Park ef al., 1988):

a ( N ) = a,

+ 1 +a,N / N ,

consisting of an unsaturable part a, and a saturable part a,, where N , is the saturation value of the photocarrier density. Writing 6 a ( N ) = a ( 0 ) - a ( N ) = z s ( N / N s ) / ( l + N / N s ) , experimental data can be most easily fit to the saturation form if 1/6a is plotted vs 1/N. Then the intercept gives l/a, and the saturation carrier density can be determined from the slope. The refractive index has a similiar form:

n(N) = n,

+ 1 +n,N / N ,

The photocarrier density N is related to the incident intensity I by the absorption and the recombination time of the carriers, T: N = alr/hv

(3)

where hv is the photon energy. The time 7 is determined by the joint probability of electrons and holes being available for recombination. Therefore, if carrier transport can be used to increase t, then the required intensity I decreases, since the scaling is proportional to IT. If 7 can be controlled through carrier transport, then the nonlinearity can be optimized to achieve as much sensitivity as possible for a given required response time. When experimental data are given in terms of the intensity dependence of the absorption change, the saturated absorption change, a,, can be most easily found by plotting log(6a) vs. log(1) and extrapolating to large log(/). The saturation intensity occurs when the absorption change reaches half its saturation value, and can be found from the value at which log(&) = log(a,/2) = log(a,) - 0.3. The use of carrier transport to control recombination time by separating photogenerated electrons and holes in semiconductors can be understood with reference to Fig.2(a). An undoped quantum well (QW) sits in an n region, surrounded by the tilted bands from two back-to-back p-i-n junctions, as shown. The tilt of the conduction and valence bands (C.B. and V.B., respectively) is created by the equalization of the Fermi levels. The undoped QW has a smaller band gap than the rest of the material, and its resonant

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS I I

II I

I I

I

I

I

I

I

iQWI

I I

I

I

59

I I

I

I I

f n

I

I

I

I

FIG.2. Semiconductor junction structures for understanding carrier transport nonlinearities: (a) QW in the n region, utilizing enhanced lifetime due to removal of holes; (b) Q W in the i region, utilizing field-moderated changes in absorption and refractive index.

absorption creates photocarriers in the QW. The electrons remain trapped in the well, but the holes tunnel out and float up to the p-doped region. The resulting local concentration of electrons in the Q W causes the available states to fill, as discussed in Chap. 1, whereas removal of the holes means that recombination times become very long. This results in very sensitive state-filling nonlinearities, although at the sacrifice of much slower response times, which can be lengthened from nanoseconds to milliseconds, decreasing saturation intensities from kilowatts to milliwatts.

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ELSAGARMIRE

Since state filling depends on the electron density, and each photon creates a single electron that lasts until it recombines, there is a direct tradeoff between sensitivity (defined by the required intensity to fill available states) and reponse time. In fact, for times less than the recombination time, the fluence F at the excitonic resonance wavelength required to cause a given change in absorption or refractive index would be the same as in undoped material, since 17 in Eq. 3 becomes F.

2. NONLOCAL NONLINEARIT~ES Carrier transport in the presence of static fields causes electrons and holes to move out of their local region, with the resultant charge separation screening the static field. This nonlocal self-modulation of the electric field results in a large nonlocal change in refractive index and/or absorption near the band edge. The concept of a carrier transport nonlinearity due to self-modulation of a static field can be understood with reference to Fig.2(b). This figure shows a quantum well (QW) in an undoped intrinsic i region subject to a static electric field. As an example, we consider a p-i-n junction whose built-in electric field comes from the tilt of the conduction and valence bands (C.B. and V.B., respectively) that is created by the equalization of the Fermi levels. The Q W acts as a resonant absorber in the static electric field. When light is incident at a wavelength near the QW resonance, both electrons and holes are photogenerated in the QW. In the presence of the static electric field, the photocarriers leave the QW by thermionic emission and/or tunneling. Free electrons will fall down to the n region, and free holes will float up to the p region. The separation of these charges introduces a screening field that reduces the static electric field from its value in the dark. The band tilt is decreased, and there is a net optically generated forward bias resulting from the photogenerated carriers. Since absorption and the refractive index in semiconductors are functions of static field, the presence of light will change the absorption and the refractive index, which is, by definition, an optical nonlinearity. Because the photogenerated electrons and holes separate in physical space, they cannot easily recombine. This lengthens their recombination times from nanoseconds to milliseconds, meaning that self-modulation will occur with very small intensities, as low as microwatts. One reason for interest in the self-modulation nonlinearities is that under optimal circumstances their figure of merit can be larger than that obtained by state filling, the localized nonlinearity seen in undoped material.

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ELSAGARMIRE

measured overall absorption decrease was (6a) = 1500cm-' with 40mW pump power, slightly higher than observed in a type I1 hetero-nipi (discussed below), giving a 40% transmission change for their 2.64-pm-thick sample. From this can be inferred a photoinduced QW-only absorption change of 6a = 1.2 pm-'. These results confirm that type I hetero-nipi's can have large enough absorption changes to make effective optically-addressed FabryPer0t SLMs. While these structures are impractically long-lived (ms) and lateral diffusion reduces their usefulness in a spatial light modulator, pixelation can remove diffusion and encourage recombination, decreasing the response times (but also increasing the power necessary to operate the device). Thus the type I hetero-nipi offers an ability to trade off recombination time for sensitivity in semiconductor nonlinearities and provides considerable increase in average absorption change over that measured in nipi's. Type 11 hetero-nipi's are designed to optimize the use of field-dependent effects, inserting the lower-band-gap material into the i regions, and are discussed next.

2. ENHANCEDNONLINEARITIS BASEDON OF INTERNAL FIELDS a.

PHOTOMODULATION

Type I1 Hetero-nipi's Utilizing the Franz-Keldysh Efect

When the i regions of a nipi structure have a smaller band gap than the doped regions, light will be absorbed predominantly in the i regions, where the internal field is located. Then the separation of photocarriers generated in this region creates a screening field that opposes the internal field. This creates a self-modulationthat can be observed at the band edge of the i region material. The Franz-Keldysh effect creates field-dependent changes in absorption and refractive index; the absorption of pump light creates a change in the absorption measured by the probe. Experimental results in such a sample are very similar to those predicted by the Franz-Keldysh analysis (Danner el a!., 1988; Garmire et a/., 1989). An overall absorption decrease of ( 6 a ) = 150cm-' was measured with 280mW/cm2 input intensity, a fractional change in transmission of only a few percent. The geometry consisted of 270-nm i regions sandwiched between 27-nm doped AlGaAs layers acting as transparent electrodes. While the results clearly demonstrate the origin of self-modulation nonlinearities, the small values measured were due to the long periodicity in the nipi, which resulted in small internal fields. Absorption due to the Franz-Keldysh effect in the i regions also can be observed in a pure GaAs nipi, as pointed out in the last section (Kobayashi, 1988; Ruden, 1989).

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ELSAGARMIRX

A sensitivity figure of merit can be defined in terms of the required intensity to achieve a given absorption change, 6a. At small fluences da(N) = a , N / N , ; since N = a+IT/hv, then k / h v = (6a/a+)(N,/a,). The required intensity decreases if the following figure of merit is increased: os = a,/N,. Note that cr, = da/N, so it can be found from the slope of 6a with N. Similarly, define a refractive sensitivity figure of merit qs = 6n/N = “SINS.

4. CHAPTER OUTLINE

This chapter will emphasize materials and devices that use self-modulation nonlinearities (Garmire et a/., 1989), with some discussion of carrier transport-enhanced state-filling devices. The self-modulation nonlinearites utilize carrier transport to turn the resonant x2 nonlinearity (electroabsorption, electrorefraction) into a resonant x3 nonlinearity (optically induced change in absorption and/or refractive index). (This is to be distinguished from the recently introduced nonresonant cascading nonlinearity.) The emphasis here will be on carrier transport nonlinearities that utilize intrinsic material containing internal fields, possibly enhanced by an external field, as opposed to the hybrid configuration, referred to as self-electmoptic efect deoices (SEEDS), in which an external circuit is an integral part of the nonlocalized nonlinearity (Lentine and Miller, 1993). Another version of self-modulation nonlinearities is photorefractivity, discussed in Chap. 5. I will first describe experimental results of carrier transport optical nonlinearities in semiconductors, including n-i-p-i and single-junction structures. I will then characterize the field dependence of absorption and refractive index at normal incidence, such as used in electrically driven spatial light modulators.This study is designed to predict the potential for such structuresin self-modulation nonlinearities. Following this I consider how these structures are used in optical experiments, exploring various device geometries and looking at some typical examples, particularly reflective Fabry-Perot etalons. I end this chapter by analyzing characteristics of the optical nonlinearities. Several features stand out. Quantum confinement of carriers substantially increases the magnitude of the nonlocal nonlinearities. Charge separation causes a serious lengthening of the recombination time, slowing down decay time into the millisecond regime. Perhaps most interesting is that recombination is an exponential function of photoexcitation level, so it can take 12 orders of magnitude, in principle, to fully saturate the nonlinearity. Before this, however, bands become flat enough that carriers become “stuck” in the QWs and cannot screen the remaining field, leading to a saturated value of the internal field.

2 OPTICAL NONLINEARITLES I N SEMICONDUCTORS

63

While carrier transport can render devices very sensitive, the long carrier lifetimes also enable substantial transverse diffusion; carriers can move laterally by as much as a centimeter before they recombine. Pixelation of the samples can eliminate lateral diffusion and can, if desired, reduce the recombination lifetime to microseconds or even sub-nanoseconds. This will enable spatial light modulator applications, which are realistic goals using the sensitivity and performance of reported devices.

11. Experimental Results on Optical Nonlinearities Influenced

by Carrier Transport This chapter focuses on planar epitaxial semiconductors in the presence of a static electric field directed normal to the plane of the epitaxial layers. Photogenerated carriers move to screen this field, forming sheets of separated charge. This photocarrier transport can influence optical nonlinearities in two ways. First, charge separation reduces carrier recombination rates so that state filling can occur more readily. State filling is evidenced as a reduction in absorption due to the presence of electrons in the conduction band; because of the Pauli exclusion principle, photoelectrons can fill available states, blocking them from additional absorption. State filling is possible at lower incident intensities if carrier recombination rates are reduced, this is done by using separation of photo-charges in internal fields. The result is increased sensitivity of the state-filling optical nonlinearity. Second, screening of internal fields by photocarrier transport causes a change in absorption and refractive index, resulting in a selfmodulation near the band edge in semiconductors. In some cases the self-modulation nonlinearities may have figures of merit larger than those observed in state filling. This section will discuss experimental results on both classes of such nonlinearities, for bulk and for quantum well (QW) devices. The geometry under consideration (see Fig. 1) is used for spatial light modulation applications. Epitaxial layers are grown and doped to enable static electric fields in the growth direction (z), sometimes with internal junction fields and sometimes with fields created by an external voltage. This section considers examples of both single junctions and multiple junctions grown one on top of the other. When several junctions alternate, they are called nipi or hetero-nipi structures. When QWs are inserted into nipi structures, they can be placed in the doped regions (called type I) or in the i regions (called type I I ) . Both types exhibit optical nonlinearities characterized by carrier transport.

64

ELSAGARMIRE

A number of experimental examples will be given here; the list is not inclusive but representative. In an effort to reduce the number of references and to concentrate on practical applications, measurements at room temperature will be emphasized.

1.

ENHANCED NONLINEARITIES BASEDON STATE FILLING WITH DECREASED CARRIER RECOMBINATIONRATES

a.

Depletion Region Eiectroabsorption Modulator

Photocharge separation can occur when a potential well traps one carrier type only. The resulting reduction in the joint density of states reduces the recombination rate, enhancing the sensitivity of state-filling nonlinearities. A simple example of this type of optical nonlinearity uses a metal-semiconductor junction (Jokerst and Garmire, 1989; Garmire et al., 1989). A nominally n-type InGaAsP layer, with band gap near 1.06pm,was grown on a semi-insulating InP substrate and covered by a gold Schottky barrier. The band diagram (Fig. 3) indicates that band bending will confine electrons

InGaAsP

InP Semi-

,epitaxial . laver

FIG. 3. Band diagram for a hetero-Schottky structure that confines electrons and not holes. The InGaAsP narrow gap n - region is designed to have its band gap at the wavelength of the Nd:YAG laser (1.06pm). and the semi-insulating InP substrate is transparent. Band bending occurs at the substrate and at the gold Schottky barrier. Photoelectrons are confined, while photo-holes tunnel out. The measured absorption decreased by 50% with 4 W/cm2 incident light intensity. (After Garmire. 1994.)

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

65

and not holes. The internal fields at the Schottky barrier and at the interface with the semi-insulating substrate were not large enough to create a significant Franz-Keldysh effect but were large enough to confine electrons while allowing holes to float up and escape, enabling sensitive band filling. This was the first normal-incidence optically addressed spatial light modulator for 1.06-pm lasers. Experiments were performed with a NdYAG laser, incident through the transparent InP substrate and reflecting off the gold Schottky barrier. Nonlinear transmission through a double pass of the epitaxial layer demonstrated an optically induced absorption reduction of up to 50% for a figure of merit of Ga/a- = 1. The observed saturation intensity (defined as that needed to reduce the absorption by a factor of 2) was approximately linearly proportional to the thickness of the absorbing region, 1 W/cm2/pm. The thickness dependence comes about because the electron-hole recombination time depends on their separation distance. Measurements of carrier recombination time in an epitaxial layer 3 pm thick yielded T = 4.2 ps. The same sample exhibited a saturation intensity I , = 4 W/cm2 and a measured low-intensity loss in the epilayer of a- = 0.4pm-'. This can be compared with measurements of undoped GaAs bulk nonlinearities by noting that al,T = constant. Typical numbers for band-filling saturation intensities are a thousand times higher, with lifetimes a thousand times shorter. This optically addressed modulator shows that providing suitable band bending to remove holes from the photoinduced sea of electrons can provide a thousand-fold reduction in the intensity needed for band filling. Of course, this carrier transport mechanism does not lower the total photon flux needed to create a nonlinearity; it merely slows down photocarrier removal and makes the device more sensitive. This approach is valuable for quasicontinuous-wave (CW) applications, in which optimized sensitivity occurs when the recombination rate is chosen comparable with the desired operation rate. A multiple-junction device offers more band curvature within a given sample length and is discussed next.

-

b. nipi Structures

The original nipi structures alternated epitaxial growth of n and p layers; depletion of carriers caused the regions between each n and p layer to be intrinsic. The term nipi (or n-i-p-i) can be considered as synonymous with modulation doping superlattice (Fig. 4). The periodic space-charge potential of the ionized impurities makes possible very efficient spatial separation of electrons and holes and provides an indirect gap in real space.

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ELSA GARMIRE

FIG.4. Band diagram for a nipi structure. The valleys are the n regions, and the hills are the p regions. Absorption can take place anywhere, promoting electrons to the conduction band (C.B.)and leaving behind holes in the valence band (V.B.). (a) Free carriers will drift to their point of lowest energy. (b) After strong illumination, the separation of electrons and holes reduces the amount of band bending. (After Danner ef al., 1988.)

The most extensive work on doping superlattices (“n-i-p-i crystals”) has been the theoretical (Dohler, 1972a, 1972b; Dohler, 1979) and experimental (Dohler et al., 1981) work of G. H. Dohler and colleagues in Germany. This group developed a method of providing lateral doping-selective ohmic contacts to the two doped regions (Sn and Sn/Zn) and demonstrated tunable band gaps by changing the potential difference between the n and p regions (Pfloog et al., 1981; Kunzel, 1982). An extensive paper in 1982 described the absorption in a 40-nm doping superlattice and discussed its tunability, both electrically and optically (Dohler et al., 1982). This paper pointed out that the consequence of the doping superlattice is an “indirect gap in real space,” in which electrons and holes collect in different physical locations. The result is that electron-hole recombination times may exceed the corresponding values of unmodulated bulk material by many orders of magnitude because the spatial overlap of the electron and hole states is small. This paper also described how photoexcited carriers collecting in the n and p regions cause a change in the potential drop between these two regions. This was seen as a good way to modulate the absorption with changing carrier concentration.

2

OPTICAL

NONLINEARITIES IN SEMICONDUCTORS

67

Considerable research on the optical properties of doping superlattices, both experimental and theoretical, was carried out in subsequent years. An excellent review paper in 1986 (Dohler, 1986) provides considerable detail and a complete reference list on early work. Dohler points out the extremely long excess carrier lifetime that results from the spatial separation of electrons and holes and indicates that this lifetime should depend exponentially on the amount of band bending. Figure 4 gives an idea of what happens when light is incident on a nipi structure with a wavelength near the semiconductor band gap. At low optical excitation, most of the structure is under the influence of a strong internal electric field. This field will alter the shape of the electron and hole wavefunctions and will correspondingly affect the absorption (Franz-Keldysh effect, Section 111). While absorption changes occur both above and below the band edge, this absorption is particularly noteworthy for its field-induced band tail. When the optical excitation increases, photogenerated electron-hole pairs will separate into their respective doped majority-carrier regions and will induce an electric diplole moment that will counteract the built-in electric field. Because the internal electric field is reduced, the band-tail absorption will tend toward that of undoped materials. That is, the absorption below the band edge will decrease with increased photoexcitation. There is, in addition, a second source of photomoderated absorption in nipi's. As carriers collect in the doped regions, they will fill available states, blocking further absorption. The result can be thought of as a photoinduced shift of the band edge to shorter wavelengths. Thus state filling in the doped regions causes a decrease in the absorption above the band edge. This may add to or subtract from absorption changes due to the Franz-Keldysh effect, depending on the specific geometry of the nipi structure. Experimental demonstration using absorbed optical intensity to modulate a second light beam in nipi structures was reported in 1986. Measurements from low temperature up to room temperature (Simpson et al., 1986; Simpson et al., 1988) compared 100- and 56-nm-period superlattices, designed with confining well depth of 1.4 and 0.6eV, respectively. Much larger effects at room temperature were observed with the 100-nm-period superlattice, a maximum of approximately 6% modulation at room temperature with an input intensity of 1.5 W/cm2 in a superlattice l p m thick; this corresponds to an absorption decrease (averaged over the entire sample length) of (6a) = 600 cm- Measurements were made at normal incidence without removal of the absorbing substrate, so only the photomodulated band tail was observed. The same paper reported modulation depth as a function of chopping frequency, finding a 3-dB roll-off frequency of 150Hz (measurements at

'.

68

ELSAGARMIRE

14 K, with 1 mW/cm* illumination), for a lifetime of 10ms. With stronger light signals, the bands are less bent, and recombination times are shorter; at 0.1 W/cm2, for example, roll-off at 2000 Hz was measured. The data can be plotted to show a linear dependence of the log of the 3-dB frequency response on the log of the incident intensity. Such nipi structures have practical limitations because of low modulation depth and exceptionally long carrier lifetime. One way around this would be to use waveguides to enhance the interaction length and selective electrical contacts to speed up the response by removing photocarriers. The selective contacts would reduce the illumination dependence of the response time. Measurements of room-temperature nonlinear absorption (with the substrate removed) through a nipi structure with a period of 80nm (4 x 10'8cm-3 doping density) provided spectra above the band edge and confirmed that the modulation depth in thin nipi films was too small to be practical (Danner et al., 1988). A pump laser induced a decrease in absorption centered around 25 meV below the band edge ( ( 6 ~ )= 350cm-' for 750 mW/cmz incident), presumably the Franz-Keldysh effect. Above the band gap a photoinduced absorption decrease was reported for the first time, appearing as a separate, larger peak centered 150 meV above the band ( ( 6 a ) = -500cm-' for 750mW/cm2). This decrease is attributed to band filling at the bottom of the n-region conduction band. The dependence of 6cr on intensity was studied for each of these peaks; the data can be replotted to use extrapolation to find both the saturation value of the absorption change and the saturation intensity. As shown in Section V, this is done by plotting log(6a) vs log (I). The plot of data from the long-wavelength peak in absorption change shows a strong saturation, extrapolating to (6a,) = 468 em-'. The saturation intensity, occurring when the absorption change reaches half its saturation value, can be found from the value at which log(6a) = log(6us/2) = log(Cra,) - 0.3. From the data, I, = 209 mW/cmz. The relation of optically induced absorption change to recombination time was measured for a fixed intensity over a wide temperature range (Kobayashi et al., 1986%1986b Kobayashi et al., 1988), demonstrating that bar = constant, as expected from Eq. (3). For an InP doping superlattice with a 400-nm period, the room-temperature lifetime at 200 mW/cmz pump was measured to be lops. Spectral measurements in GaAs doping superlattices with periodicity of 260nm could be interpreted on the basis of a semiclassical model that included the Franz-Keldysh effect (see Section 111) both below and above the bulk GaAs band gap, as well as band filling and band-gap shrinkage (Ruden et al., 1989). Their simulation contained the same features as the measured transmission change. A long-wavelength absorption decrease as

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

69

large as (601) = 580cm- was measured; in a 2.6-pm-thick sample, this gave a 15% fractional transmission increase. A plot of GaL vs. log I (Fig. 4 in their paper) has the same shape as modeled in Fig. 26, and is approximately linear over the mid-range of intensities. A plot of log(6aL) vs. log1 can be estimated to extrapolate to 20% higher than observed at 7 W/cmz, with a saturation intensity of 1.6 W/cmz. This paper also reported measurements on a short-period nipi (80-nm period) of a geometry very similar to that reported by Danner. Measurements in the high-excitation regime (at 50 times the power levels used by Danner) showed a much smaller long-wavelength absorption decrease, only ( 6 a) = 37 cm-l. This was accompanied, however, by a photoinduced absorption increase of (6a) = 200cm-', which was explained by a shortwavelength photo-induced band-gap shrinkage. In order to determine a figure of merit, it is necessary to know the baseline absorption as well as the photoinduced absorption change. In experiments on an 80-nm-period superlattice (Tobin and Bruno, 1991), the maximum long-wavelength value of the figure of merit was &/a- = 0.23 at 1.38 eV. This is small because the absorption data show a long tail resulting from the Franz-Keldysh effect in the nipi structures and the photoinduced absorption changes are not large enough to reduce this tail significantly, at least near the band edge. This paper also describes results on a nipi structure grown with a combination of doping densities and layer thicknesses that theoretically leads to a spatially indirect effective band gap of OeV so that it was expected to be semimetallic. The background absorption in this sample was approximately two times lower, with the same photoinduced absorption change, increasing the figure of merit by a factor of 2. In direct-band semiconductors, modulation doping (nipi structures) gives rise to some unique optical properties due to quantization of energy levels, particularly at low temperatures. However, the room-temperature nonlinearities tend to be small, and they sit on a lossy band tail, so would be useful only in waveguides; improved performance as a nonlinear optical material can be achieved by inserting QWs within the modulation-doped structure. c.

Type I Hetero-nipi Structures

The incorporation of a heterostructure into a doping superlattice, resulting in a "hetero-nipi crystal," was originally suggested (Dohler, 1981; 1983) to provide, within the middle of the doped regions, a central layer of undoped, narrower-gap material. The original purpose was to spatially isolate the carriers from the doping impurity centers, thereby increasing

70

ELSAGARMIRE

mobility. Later it was realized that quantum confinement of these carriers would enhance their optical absorption properties. The incorporation of narrow-band-gap material within the doped layers was designated as type I hetero-nipi, which provides optical nonlinearities through state filling. After realization that placing the narrow-band-gap materials in the i regions also could be useful, these were designated as type I1 hetero-nipi. The latter enable self-modulation nonlinearities and operate on photoinduced internal field screening. Two reviews of hetero-nipi results were published in 1990 (Dohler, 1990; Kost, 1990). This section reviews optical measurements on such structures. A typical example of a type I hetero-nipi is shown in Fig. 5. When used for nonlinear optical applications, light incident at the exciton resonance energy hv, is absorbed only in the QW, whose levels are below the band-gap of the rest of the nipi structure. Majority photocarriers (here electrons) stay in the wells, but minority carriers (here holes) tunnel out and drift to their lowest energy potential. The buildup of only electrons in the wells causes the state-filling nonlinearity at very low optical intensities. In addition, as holes float out of the wells, they build up a screening field that reduces the internal field in the i layers, which can cause photoinduced absorption change due to the Franz-Keldysh effect. An alternative approach uses a two-wavelength pump-probe experiment to pump the type I hetero-nipi with a large enough photon energy hv, that it can be absorbed anywhere in the nipi. This liberates electrons that can drop down into the wells, providing an absorp-

\ P - * I I

I

h",

I

I

B. FIG. 5. Band diagram for a type I hetero-nipi showing photoexcitation either at the excitonic resonance hv, or with an above-band pump hv,. In either case, the photoelectrons collect in the quantum wells, while the holes float away. The nipi periodicity is A. (After Ianelli et a/., 1989.)

2 OPTICAL NONLINEARITLES I N SEMICONDUCTORS

71

tion change near the exciton resonance that is observed by the probe. The first heterodoping nonlinearity did not use QWs, but a narrow-gap n-layer 110 nm wide consisting of InGaAsP. This was surrounded by 110-nm wide-gap p-InP layers (Kobayashi et al., 1988); 20 pairs were grown with periodicity A = 220 nm. The narrow-gap n regions absorbed the light; holes escaped to the p regions, creating a screening field that reduced the band bending. The result was a combination of band filling (made more sensitive by the reduced carrier recombination in the doping superlattice) and the Franz-Keldysh effect in the depleted regions at the pn interfaces. This structure is similar in concept to the type I hetero-nipi introduced by Dohler. The researchers observed strong below-band-gap absorption that did not depend on temperature, which they attributed to the Franz-Keldysh effect from the internal fields within the n layers. They modeled the measured nonlinearity by assuming that the absorption change below the band edge was due to the Franz-Keldysh effect, while above the band edge it was due to band filling, and they obtained reasonable agreement with experiment. The maximum ( 6 a ) at room temperature was 140cm-'; the long-wavelength figure of merit 6a/a- x 0.15%, comparable to the pure nipi and too small for most practical applications. Increased modulation depths in type I hetero-nipi's are achieved by using QWs in the doped regions to enhance the size of the nonlinear modulation. This occurs because quantum confinement increases the excitonic resonance, enhancing the available absorpfion change. Researchers at Jet Propulsion Laboratories reported modulation in a periodically doped GaAs structure with a strained, narrow-gap InGaAs quantum well in each of 20 n-doped regions (Iannelli, et al., 1989). The structure had an undoped QW 12nm thick surrounded by n+ doped layers, 35nm on each side, and each was adjacent to p + layers 70nm thick, for a nipi period A = 152 nm. The maximum modulation depth was an increase in transmission from 31% to 34% at 5 mW of pump power, with an effective recombination time of about 1 ms. The absorption loss in the on state was 12%, giving a maximum figure of merit of &/a- = 0.8. The QW-only absorption decrease was 6a = 0.42pm-'; averaged over the sample, this gives (601) = 340cm-' because of the 8% Q W filling factor. The geometry was designed for optically addressed spatial light modulator applications in which the pump light (write beam) is at a wavelength shorter than the band gap so that the write beam can be absorbed throughout the nipi structure. The photogenerated free carriers then fall into the QW, where they alter the absorption enough to be measured by a probe beam (write beam) set at the wavelength of the QW absorption. Further results were reported using delta doping (Larsson and Maserjian, 1991b) in the geometry shown in Fig. 6. Doping was inserted in monolayer

72

ELSA GARMIRE

InGaAs QW

FIG.6. Energy band diagram for delta-doped type I hetero-nipi showing the illumination that causes the state-filling absorption change and indicating the hole transport that leads to long lifetimes and enhanced sensitivity. (After Larsson and Masejian, 1991b.)

planes, providing more uniform fields inside the nipi structure. The QWs were InGaAs 6.5nm wide, with n-doping planes inserted on either side of the QW. The p-doping plane was inserted midway through the 78-nm GaAs barriers that separated the QWs; the nipi periodicity was A = 85 nm. The Q W filling factor was the same (8%); although the wells were twice as small ifor larger quantum confinement), the period was half, allowing for twice the number of QWs per unit length. The nipi sample had 44 wells, for a total thickness of 3.8 Vm. Measurements of the absorption spectrum for various optical pump powers are shown in Fig. 7. Both the absolute absorption and the fractional absorption change are shown. The single peak in the absorption change indicates a pure absorption decrease at the resonance and no wavelength shift. Such data are typical for pump-probe experiments in type I hetero-nipi’s. The peak absorption (with no pump) was a+ = 1.65 p m - ’ (QW-only), decreasing to a - = 0.75 pm-’ as the intensity increased from 0 to 100mW/ cm’, for an absorption change of 6a = 0.9 pm- ((6a) = 720 cm- I ) and a figure of merit /? = &/a+ = 0.58, with lifetimes in the millisecond range. This result can be compared with undoped samples to confirm that there is a direct tradeoff between sensitivity and lifetime. The decrease in sensitivity, from 25 kW/cm2 typically required to saturate the absorption in undoped QWs, to a measured value of 100 mW/cm’, is comparable to 250,000 times lengthening of lifetime, from the 10 ns typically observed in undoped material to the 2.5ms typical for nipi structures. In order to predict

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

€

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zt 66

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tl 40

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1100

FIG. 7. Experimental results of the type I hetero-nipi shown in Fig. 6. Absorption spectra are shown for various powers of excitation P,, in a sample area of 0.1 cm'. Increasing power causes decreasing absorption due to state filling. The fractional absorption change is also shown. (After Larsson and Masejian, 1991b.)

performance in an impedance-matched Fabry-Perot (IMFP), the figure of merit should be extrapolated to large intensity by plotting the log@) vs log (I). The result shows 8, = 0.8. Inserting this value into Eq. 4 (derived in Section IV) predicts that the reflectivity can be varied from 0 to 44%. This type I nipi was promising enough that it was used in an asymmetric Fabry-Perot to operate as an optically addressed spatial light modulator (SLM), which will be discussed in Section V. Type I hetero-nipi's were reported in a GaAs/Al,.,Ga,.,As structure (Hibbs-Brenner et al., 1994); a single period consisted of a 10-nm undoped GaAs Q W surrounded by 10-nm AlGaAs n regions. Adjacent to the n layer was 15 nm of undoped AlGaAs, followed by 20 nm of p-AlGaAs and another 15 nm of undoped AlGaAs, for a periodicity A = 80nm. The reported structure had a total of 33 QWs and a QW filling factor of 12.5%. The

74

ELSAGARMIRE

measured overall absorption decrease was (6a) = 1500cm-' with 40mW pump power, slightly higher than observed in a type I1 hetero-nipi (discussed below), giving a 40% transmission change for their 2.64-pm-thick sample. From this can be inferred a photoinduced QW-only absorption change of 6a = 1.2 pm-'. These results confirm that type I hetero-nipi's can have large enough absorption changes to make effective optically-addressed FabryPer0t SLMs. While these structures are impractically long-lived (ms) and lateral diffusion reduces their usefulness in a spatial light modulator, pixelation can remove diffusion and encourage recombination, decreasing the response times (but also increasing the power necessary to operate the device). Thus the type I hetero-nipi offers an ability to trade off recombination time for sensitivity in semiconductor nonlinearities and provides considerable increase in average absorption change over that measured in nipi's. Type 11 hetero-nipi's are designed to optimize the use of field-dependent effects, inserting the lower-band-gap material into the i regions, and are discussed next.

2. ENHANCEDNONLINEARITIS BASEDON OF INTERNAL FIELDS a.

PHOTOMODULATION

Type I1 Hetero-nipi's Utilizing the Franz-Keldysh Efect

When the i regions of a nipi structure have a smaller band gap than the doped regions, light will be absorbed predominantly in the i regions, where the internal field is located. Then the separation of photocarriers generated in this region creates a screening field that opposes the internal field. This creates a self-modulationthat can be observed at the band edge of the i region material. The Franz-Keldysh effect creates field-dependent changes in absorption and refractive index; the absorption of pump light creates a change in the absorption measured by the probe. Experimental results in such a sample are very similar to those predicted by the Franz-Keldysh analysis (Danner el a!., 1988; Garmire et a/., 1989). An overall absorption decrease of ( 6 a ) = 150cm-' was measured with 280mW/cm2 input intensity, a fractional change in transmission of only a few percent. The geometry consisted of 270-nm i regions sandwiched between 27-nm doped AlGaAs layers acting as transparent electrodes. While the results clearly demonstrate the origin of self-modulation nonlinearities, the small values measured were due to the long periodicity in the nipi, which resulted in small internal fields. Absorption due to the Franz-Keldysh effect in the i regions also can be observed in a pure GaAs nipi, as pointed out in the last section (Kobayashi, 1988; Ruden, 1989).

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

75

A smaller thickness between the n and p regions gives larger absorption modulation because of higher internal fields. When 75-nm GaAs i layers were sandwiched between 13-nm doped AlGaAs layers, photomodulation created a maximum overall absorption decrease of (6a) = 1300cm- with 40-mW pumping (Hibbs-Brenner, et al., 1994). The internal field is calculated to be 3.6 times larger than in the previous structure; the 8.7-fold increase in measured absorption modulation is reasonable because the Franz-Keldysh effect depends quadratically on field.

b. Type II Quantum Well Hetero-nipi Structures When quantum wells are placed in the i regions, the nonlinearities arise from self-modulationof the internal fields and can be very large (Kost et al., 1988a). The band diagram for the first such structure demonstrated experimentally is shown in Fig. 8. In this case, QWs are inserted in the i regions so that photocarrier charge separation induces a screening field that modulates the quantum confined Stark efect (QCSE). This is the name given to the electric field dependence of the absorption and refractive index in QWs, as discussed in Section 111. A static field across a QW creates

4 200

H 400

FIG. 8. Energy band diagram for a type I1 hetero-nipi showing quantum wells in the i regions that preferentially absorb light, liberating free carriers that move to their respective majority carrier doped regions. This separation of charge partially screens the field E, causing the quantum well absorption to change. (After Garmire et al., 1989a.)

76

ELSAGARMIRE

increased absorption below the band edge. Since the nipi field will be larger In the dark than under illumination, absorption below the band edge will decrease as the bands flatten due to the screening field built up from photocarriers. Thus, applying pump light will decrease the absorption below the band edge, the same sign as seen in the saturable absorption that results from state filling. However, because the quantum well exciton regains its oscillator strength as the field is removed, the absorption will increase at photon energies above the excitonic resonance, rather than decrease. This is seen in Fig.9. The self-modulation nonlinearity measured in the type I1 hetero-nipi structure shown in Fig.8 appeared to saturate to a maximum value of & I = 0.25 pm- (QW-only) measured at 50 mW/cm* (Garmire et al., 1989), for an overall fractional transmission increase of approximately 12%. The MOCVD-grown structure incorporated 8 wells per i region, with a total QW filling factor of 35%, much larger than the type I M Q W hetero-nipi’s reported above. The wells were 10nm, separated by 10-nm AlGaAs barriers. The 40-nm doped regions were each surrounded by 20-nm spacer layers to ensure homogeneous fields across the QW. The i regions were 190 nm thick, and the overall period of the nipi structure was 460 nm, so only three periods could be incorporated in the 1.38-pm-thick sample. While the absorption change in the Q W due to QCSE is considerably smaller than in the type I nipi, the filling factor is much larger, so the overall transmission change is comparable. While GL could be estimated only, it

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77

2 OPTICAL NONLINFARITIES IN SEMICONDUCTORS

appears that at the wavelength region where maximum 6a was measured, a- (under illumination) was 0.4pm-' (Garmire et al., 1989), so that f = Ga/a- = 0.62. Larger absorption modulation is required before this

nonlinearity can be practical. A similar structure, grown by MBE (Ando et al., 1989a), consisted of five periods containing ten 8-nm wells (and 4-nm barriers) in each i region, for a total of 100 wells. The doped regions were each 50nm and were surrounded by 40-nm spacer i layers so that each period was 500nm. The 32% filling factor was comparable with that of Kost and colleagues. The result of narrower excitonic resonances and narrower wells (plus some Fabry-Perot enhancement) was that twice as large absorption change was reported, 6a = 5OOOcm-'. With 100 QWs in a sample 2.5 pm thick, this implies a fractional transmission change 6T/T = 40%. The data also indicate a- x 7000cm-', so that f = 6a/a- = 0.7, comparable to the preceding case and a factor of two less than the type I hetero-nipi figure of merit (for which f = 1.4). Residual loss must be reduced before the type I1 hetero-nipi can become practical. Two-pass reflection measurements were made on similar MQW structures grown on Bragg mirrors (Law et al., 1989), which have the advantage that the substrate need not be removed. This structure had a higher internal field (more than twice as large), due to smaller separation between n and p regions (71 nm), with a 160-nm period. This made it possible to observe a larger optically modulated QCSE. The authors reported 6a = 7500cm-I using 7.9-nm wells in a structure with 39% filling factor. In two passes of the epitaxial layer, the output signal increased from 40% of the probe beam to 70% in the presence of a 2 mW optical control beam. These results were promising enough that type I1 hetero-nipi's have been used in Fabry-Perot etalons for optically addressed spatial light modulators (Section V). Recent measurements in our laboratory have found similar numbers (Yang et al., 1995). With 20 i layers, each 124 nm thick, containing four 8-nm GaAs QWs with Al,,,Ga,.,As 4-nm barriers (including 40-nm spacer layers to achieve field uniformity across the QW)and doped regions, we observed a saturation value of 6a, = 0.71 pm-' and a saturation intensity of I, = 0.34mW. With a 16.5% confinement factor, this gives <6a) = 0.12prn-' and 45% fractional transmission change in a single pass, the largest reported to date in a type I1 hetero-nipi. Typical nonlinear absorption change spectra from this sample are shown in Fig. 10. The spectrum looks like a differential, which is the signature of a pure absorption shift, as can be seen by comparing with Fig.9. The shift in the zero crossing is a measure of the shift in the exciton resonance due to QCSE, since the field is reduced by screening. A residual absorption of 01- = 0.3 pm- was reported in Dr. Yang's Ph.D. thesis, so that f = &/a- = 2.4, better than

'

78

ELSAGARMIRE

0.4

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Wavelength (nm) FIG. 10. The spectrum of absorption change in a type I 1 hetero-nipi for various pump intensities. The differential spectrum implies a spectral shift to the exciton resonance, which can be approximated by the zero crossing. (After Yang, 1997b.)

any other reported type I1 hetero-nipi, but, since f l = 0.7, the figure of merit is still under that reported for type I hetero-nipi's. These figures of merit predict that a properly designed IMFP should have a maximum reflectivity of 30%. For longer-wavelength operation (980 nm), strained-layer InGaAs/GaAs type I1 MQW hetero-nipi's were fabricated (Larsson and Maserjian, 1991c) with a periodically delta-doped structure in order to shorten the period and further increase the internal fields. Six periods containing 4 QW (6.5 nm wells with IO-nm barriers) per i region were used, with 100nm between doping planes, resulting in a 200-nm period and a 26% filling factor for the 48 wells. A Q W absorption decrease 6a = 0.8 pm-' was measured at 100mW/cm2. For a total QW length of 0.31 pm, this implies a fractional transmission change of 25%. The reported figure of merit was /?= 6u/ a+ = 0.8, which is the same as observed in type I hetero-nipi's and predicts that when inserted into an IMFP, the maximum reflectivity would be 44%. As an alternative to the ternary InGaAs composition, QWs may be made of a short-period strained-layer InAs/GaAs superlattice. These were inserted in a type I 1 hetero-nipi structure and the nonlinear performance measured (Kost et al., 1991). The desire here was to use an epitaxial growth technique for strained materials that could be scaled to very thick layers without the buildup of strain dislocations. A t moderate power levels ( < 10W/cm2), we

2

79

OPTICAL NONLINEARITIES IN SEMICONDUCTORS

',

observed 6a = 1300 cm- presumably smaller than the ternary results because of wider wells with less indium (quantum confinement was smaller, reducing the excitonic resonance). At very high powers, additional absorption saturation occurred, which was attributed to exciton saturation and continuum state filling (described below).

c.

Coupled Quantum Wells

In an effort to improve the optical nonlinearities in type I1 hetero-nipi structures, an array of double, symmetric, coupled quantum wells (CQW) was inserted into the i layers, rather than an array of single QWs (Kost et al., 1989a). The optically pumped absorption change is shown in Fig. 11.

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Wavelength (nm) FIG. 1 1 . The spectrum of absorption change in a type I1 hetero-nipi containing coupled quantum wells in the i layers for various intensities. Note that the long-wavelengthabsorption increases with pump level, opposite the effect measured in single quantum wells. (After Kost, 1989a.)

80

ELSA GARMIRE

What is most interesting is the fact that below the band edge the absorption increases with illumination rather than decreases, a sign that is opposite that in single QWs. This is explained because reduction of the internal field increases the oscillator strength between the wells and therefore the absorption. These results are consistent with results from electrically addressed CQW modulators (Section 111). The measured peak absorption increase was 6a = 2500cm-', at a wavelength of 813 nm. This is similar to the values observed in comparable structures containing single QWs. Unfortunately, however, the optically induced absorption change takes place on top of a large background loss (almost half its value on resonance).Thus CQWs have smaller figures of merit.

3. COMBINED NONLINEARITIES a.

Type I Hetero-nipi Structures

Several efforts have been made to increase the nonlinearity by combining state filling with self-modulation of internal fields. This was first observed in a heterodoping superlattice (Kobayashi et a/., 1988), as described previously. The sum of the Franz-Keldysh effect and band filling could explain the nonlinearities that were observed. Additional studies (Ruden et al., 1989) confirmed this interpretation. More recently, electroabsorption modulators with large absorption change have been made in type I hetero-nipi's by combining band filling in the n-doped layers with the Franz-Keldysh effect in the barriers, when the design is chosen to positively combine the two effects (Gulden et al., 1994; Kneissl et al., 1994). A proper design added 1450cm-' from the FranzKeldysh effect to 12,000cm-' from band filling. A contrast ratio exceeding 2:i was observed for a 3.55-pm-thick pseudomorphic InGaAs/GaAs superlattice structure using a voltage swing of 3.3V. This nipi structure had selective contacts and was used as a modulator rather than as a nonlinear device. Nonetheless, it shows that, in principle, larger nonlinear effects can be obtained by combining nonlinear mechanisms. b.

Type 11 Hetero-nipi Structures

Combined local and carrier transport nonlinearities were observed in type I1 hetero-nipi's at high intensities (Kost et al., 1991). The apparent saturation of the QCSE self-modulation nonlinearity was overcome for intensities greater than 30 W/cm2; state filling and exciton screening contributed to the nonlinearities at these higher power levels, whereas at intensities below

2

OPTlCAL

NONLINEAR~TIES IN SEMlCONDUmORS

81

6 W/cmz there was pure self-modulation of QCSE. This led to a doubling of the transmission change and occurred because carrier transport flattened the bands, allowing photocarrier accumulation within the wells. This picture was confirmed by picosecond studies (McCallum et al., 1992) that will be described in more detail in Section V.

4.

SELF-MODULATION OF EXTERNAL FIELDS

One way to enhance the size of the optically induced absorption change is to provide an external bias to increase the field across the i layer. This can be done across a simple p-i-n (or m-i-m) junction, or across a nipi structure through use of doping-selective contacts.

a. Optical Modulation of a Single Junction The earliest considerations of optical modulation of semiconductor junction fields was in the context of the search for a practical optical bistable device. In Russia, beginning in 1982, Ryvkin and coworkers investigated self-modulation of the Franz-Keldysh effect (Ryvkin, 1981, 1985). Similar concepts were developed in this country using QCSE by Miller and coworkers beginning in 1984 (Miller et af., 1984, 1985b), and were made manifest in the self-electroopticeffect device (SEED). This device combined the QCSE modulator with optical detection within the same structure, causing an optoelectronic feedback that, when positive, gives optical bistabilty. The original SEED consisted of a photodiode containing QWs, an electrical load (resistor or another photodiode), and a reverse bias voltage supply (Fig. 12). The SEED description is usually given in terms of the photocurrent that the photodiode detects. From the point of view of carrier transport nonlinearities, the photodiode can be considered a QCSE modulator, and the photocurrent can be considered a manifestation of carrier transport. The SEED literature considers that the photocurrent changes the load line. When photocurrent is blocked from flowing, it builds up a charge on the electrodes that acts as a screening field, providing a self-modulation nonlinearity. The SEED has had a very rich development history, integrating a photodetector with an optically controlled modulator and a variety of electronics. A systematic investigation of SEED devices is beyond the scope of this chapter, and readers are referred to a recent topical issue of the Journal of Quantum Electronics (Forrest and Hinton, 1993). In the context of normal-incidence optical modulation using QWs, the first report of self-modulation in an external field was in a p i - n diode that

ELSAGARMIRE

82

n

'bias

I

P

"0

Pout,

(b)

FIG.12. Diagrams for two self-electroopticeNect device (SEED) configurations:(a) resistor biased and (b) photodiode biased. Both use an external circuit to aNect a hybrid optical nonlinearity. (After Lentine and Miller, 1993.)

contained asymmetric CQWs in the i region (Little et a/., 1987; Weber, 1989). It was observed that photoabsorption provided an effective forward bias voltage. A n optically induced reduction in the Stark shift caused an optically induced absorption decrease below the band edge. An absorbance change of between 0.3 and 0.4 was measured as the incident intensity changed from 0.1 nW/cm2 to 1 pW/cm2. The CQW structure contained an 8-nm GaAs well, a 5-nm AlGaAs tunnel barrier, a 4-nm GaAs well, and a 10-nm AlGaAs separation layer. There were 40 of these CQWs. A field of 54 kV/cm, corresponding to a 5.6-V external bias was calculated to bring the energy levels of the two wells into resonance in such a structure. While CQWs show unique features at low temperatures that can exhibit optical nonlinearities due to a polarization induced by optical pumping (Le et a/., 1988), at room temperature, measurements of electroabsorption in CQWs showed no great advantage over the quantum confined Stark effect; subsequent studies of optical modulation of single-junction devices have tended to use single QWs. Another context for investigating optical modulation of the field within an intrinsic or semi-insulating semiconductor is for photorefractive devices

2 OPTICAL NONLINEARITIES I N SEMICONDUCTORS

83

and applications. Photorefractivity in semiconductors represents another form of self-modulation of internal fields and is the topic of Chap. 5, so it will not be discussed here.

b. Modulation of Multiple Junctions with Selective Contacts

A high field can be applied across the multiple i regions in a nipi structure by using external electrical contacts, as long as these contacts can be selective, connecting only to one doping type and not to the other. Then lateral contacts, as shown in Fig. 13, can provide a means for creating a larger potential drop between the p and n doped layers than the internal field alone. In addition, these contacts allow the possibility of externally controlled nipi devices to speed up the response. To date, almost all such structures have been used for electrically driven modulators, but they have the potential to enhance optical modulation by increasing QCSE. A review of the state of this technology is given here. One of the early successful methods for doping-selective contacts used a shadow mask during growth of the epitaxial material to create a lateral variation in the composition. The masked growth resulted in pixels that contained an n-i-n-i region on one side and a p-i-p-i region on the other (Dohler et al., 1986; Chang-Hasnain, 1986). Advances in this technique using anisotropic etching of GaAs and AlGaAs layers to create the necessary

FIG. 13. Geometry for doping-selective contacts to a mesa-etched type I1 hetero-nipi, allowing it to support external fields through lateral electrical contacts. Normal incidence illumination would be from above. (After Goossen et a[., 1993.)

84

ELSAGARMIRE

shadowing during the MBE growth process, made smaller pixels possible (Gulden et al., 1993). The shadow-masking technology has led to a high-contrast electrooptic nipi doping superlattice modulator operating at normal incidence in transmission, using selective contacts (Linder et al., 1993). This electrically driven modulator uses both Franz-Keldysh absorption changes and enhancements by a carrier- and field-induced Bragg effect within the nipi crystal, resulting in a fractional transmission change of 65%, which can be described by an averaged absorption change of ( d a ) = 1900cm-'. The advantage of using the Franz-Keldysh effect is the lack of temperature dependence that is a characteristic of the QCSE. The results were explained by including an enhancement due to Bragg reflection from the nipi interfaces interfering with the Fresnel reflection at the surface air-dielectric interface. The pixels were 38 pm wide and 270 pm long and demonstrated a 6-MHz cutoff frequency, limited by the RC time constant of the laterally contacted doped layers. Predictions were that 3pm x 3pm pixels would have a 2GHz cutoff frequency and that they would be useful for opto-optical logic devices. It has been found that selective contacts also can be provided by mesa etching pixels and then evaporating AuZn on one side of each pixel and AuSn on the other (Goossen et al., 1993). AuZn provides an ohmic contact with the p-doped layers but a blocking Schottky contact with the n-doped layers. Conversely, AuSn provides an ohmic contact with the n-doped layers but a blocking Schottky contact with the p-doped layers. A normalincidence electroabsorption modulator was demonstrated in a nipi structure that incorporated MQW in the i region, grown on an integrated Bragg mirror, and operating in reflection with an antireflection coating on the top surface. By removing Fabry-Perot effects, a large operating temperature range was demonstrated (70 OC), over which the voltage-induced reflection change was at least 22%, with a maximum value of 37%. The predicted switching time for an optimized device was 4ns. The magnitude of the reflection change was smaller than expected, which was attributed to dopant redistribution into the MQW regions, giving rise to variation in the static electric field and therefore broadening the exciton. The speed of externally contacted nipi structures is limited by the resistivity of the contact layers, since current must enter and leave from the sides of the structure. A careful analysis of the tradeoffs between drive voltage and speed in a waveguide type I1 MQW hetero-nipi modulator with selective contacts (Koehler and Garmire, 1995) showed that speeds of over 300 MHz are possible in a modulator 7 pm wide and 300 pm long, while maintaining a resistance of <200Q and internal field of t 3 7 5 kV/cm, an optical loss under SO%, and no forward bias. If such constraints were dropped, the analysis showed that speeds up to 700MHz are possible. A

2 OPTICAL NONLINEARITUS IN SEMICONDUC~ORS

85

modulator based on this design, operating at 1.06pm and using InGaAs/ GaAs QWs, was demonstrated to operate with a 3-dB bandwidth as high as 110 MHz (Koehler et al., 1995, 1996). Comparable switching-speed studies were carried out at normal incidence on a type I1 MQW hetero-nipi modulator (Pfeiffer et al., 1996). It was shown that 250-MHz operation was possible for a device of 5 x 70 pm dimensions. The response time could be well fit to an equation of the form

where L and W are the dimensions of the pixel (W is the width of the electrical contact, and L is the length of the doped region over which the electrical contact must drive the current), cpnis the capacitance per unit area, pn and pp are the sheet resistivities of the doped layers, p , is the contact resistance per unit pixel length, and R , is the load resistance. (This paper used a modulator that had originally been designed for use in a waveguide configuration. In that configuration, as defined in the preceding equation, W would be the optical waveguide path length, and L would be the optical width of the modulator stripe.) Extending the preceding analysis to normalincidence pixels of dimensions 25 x 25 pm, a response time of approximately 7 ns would be predicted for a comparable n-i-p-istructure. Further work on a selective contact modulator built by the maskedgrowth technique (Huang et al., 1996) showed nonlinear optical switching of the structure at normal incidence, when its contacts were shorted, by using an ultrashort pump pulse and probing in a time long enough to ensure that the optically generated carriers had escaped the wells but short enough that negligible recombination had occurred. The authors measured an optically-induced absorption increase of 6a = 6000cm-' above the band edge and a decrease of 6a = 4000cm-' below the band edge with a fluence of 8.6 pJ/cm2. With an applied bias as low as 1 V, a comparable absorption change occurred. The structure had four periods of an interdigitated nipi structure with 5 InGaAs wells per i region, each 10 nm thick, for a total Q W thickness of 0.4pm, implying a fractional change in transmission of 24%. The nonlinear pump-probe experiments demonstrated that the opticallygenerated carriers moved to their respective doped regions, screening the internal field. At the highest fluences the absorption change spectrum was very similar to that made with a forward electrical bias of 0.9 V, enough to essentially cancel the internal field. This data confirmed that photocarrier transport induced by picosecond pulses can cancel internal fields, when the fluences are large enough.

-

86

ELSAGARMIRE

111. Field Dependence of the Optical Properties of Semiconductors

The origin of carrier transport nonlinearities lies in the change of static fields caused by the screening due to transport of photocarriers. To understand these nonlinearities, therefore, it is necessary to understand the origin of the field-dependent absorption and refractive index within semiconductors. While field-dependent absorption in bulk semiconductors has been known for a long time (the Franz-Keldysh effect), recently quantum wells have been shown to exhibit an order of magnitude larger effects (the quantum-confined Stark effect). In both cases, changes in absorption carry with them a related change in refractive index through the Kramers-Kronig relation. It is useful to have a clear understanding of field-dependent effects that can be used in self-modulation nonlinearities. These field-dependent effects are typically used in electrically driven optical modulators. This section will describe the theoretical approaches and some experimental results for both bulk semiconductors and quantum wells. 1. ABSORPTIONSPECTRA I N DIRECT-BAND SEMICONDUCTORS

The traditional approach to calculating absorption assumes that photoexcited electrons and holes are independent. However, their Coulombic attraction is crucially important to an understanding of near-band-edge absorption. To calculate absorption in a semiconductor according to the effective mass approximation, without considering Coulomb attraction between electrons and holes (Kane band theory), the two-level absorption is multiplied by the probability that the electron is in the ground state (valence band) and multiplied by the density of states. Because photons carry very little momentum, conservation of momentum requires that the initial and final states have the same momentum, making it possible to use a joint density of states. Thus the absorption in a direct-band semiconductor at photon energy hw, in the effective mass approximation, is proportional to the joint density of states (Chuang, 1995, p. 355)

where the proportionality constant Ao(hw) = 7rwIp,,12/(nrcq,), with Ipcco,l as the matrix element between the conduction and valence bands, n, as the refractive index; c and E, are the velocity of light and permitivity in free space, respectively. The joint density of states is given by

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

87

The semiconductor absorption is thus proportional to the square root of the photon energy distance from the band edge (of energy W,). Numerical evaluation is convenient using A,(ho) = (n/6)e2Wp/(nrcE,m,o), where W, is an energy parameter (in eV units) describing the transition matrix element (and is typically ~ 2 0 e V and ) rn, is the free electron mass. Equation (6) is only approximate because it ignores excitonic effects as well as the band tails that cause absorption below the band edge due to doping and impurities. The preceding analysis ignored the Coulomb attraction between the optically excited electron and hole. In fact, they experience an attractive potential given by ez/E,r that has hydrogenic-likeenergy levels involving the electron and hole effective masses me and mh, respectively. The bound exciton energy levels are given by W, = -Ry/n2, where R, is the exciton Rydberg energy, with R, = (mre4)/[2h2(4~.Q2].Also, the exciton Bohr radius is given by a, = (4x~,h~)/(rn,e~), where the reduced mass rn, is given by l/m, = l/me + l/m,,, and E, is the static dielectric constant. For GaAs, the exciton binding energy (n = 1) is 4.2meV. At room temperature, kT = 23 meV, so the excitons are thermally ionized; they can be spectroscopically resolved only at low temperatures. However, by contrast, CdS has excitons with binding energy of 29 meV that can be resolved at room temperature. In any event, the excitons change the absorption substantially from that predicted by Kane band theory, particularly near the band edge. Since this is the region of interest in nonlinear optics, it is important to realize that excitonic effects can never be ignored in semiconductor nonlinear optics. Chuang (pp. 550-552) shows that inclusion of excitonic effects and finite linewidth due to phonon and impurity scattering leads to the following equation for absorption at radian frequency o in the absence of static electric field (in three dimensions):

where the deviation from the band-gap energy (W,) is defined by

x = (ho- Wg)/Ry,with R, as the exciton Rydberg energy. Finally, y is the resonance linewidth, and S is the Sommerfeld enhancement factor:

The absorption in bulk semiconductors near the band edge can be reasonably well represented by the preceding model.

88

2.

ELSAGARMIRE

FRANZ-KELDYSH EFFECT

a. Theoretical Understanding In the presence of an electric field, the absorption in bulk semiconductors is altered near the band edge (the Franz-Keldysh efect) and is described in some detail on pages 546-549 in Chuang’s book (Chuang, 1995). The absorption increase with electric field just below the band edge can be understood by analyzing an optically excited electron in the conduction band at position z = 0. Since the electron moves in the electric field, gaining a kinetic energy eEz, it might have been possible for the electron to reach the conduction band with somewhat less than the band-gap energy. Because an electron excited into the gap has a negative energy with respect to the conduction band, the wavefunction, and therefore the probability of having an electron of energy W (and therefore the absorption a), will fall off exponentially with energy below the band-edge energy W,, so that a( W) oc exp[ -( W

- Wg)/AW ]

(10)

where A W is the characteristic Franz-Keldysh parameter, which depends on electric field. Solution of the Franz-Keldysh effect under field E , in the effective mass approximation and without Coulomb attraction, gives (Chuang, 1995)

An analytic form for the absorption, derived from Schroedinger’s equation (using the effective mass approximation), has the form of Airy’s functions and their derivatives. The resulting equation (13.2.16 in Chuang) is

where X = (ho- Wg)/AWand C = A,47t(21n,/h~)~’~. Of course, this equation does not include excitonic effects and is only approximate. Experiments have confirmed (Van Eck et a]., 1986 Calhoun and Jokerst, 1993a) that these traditional calculations based on effective mass approximation (Seraphim and Bottka, 1965; Bennett and Soret, 1987) underestimate the experimental results. A phenomenological procedure (Rees, 1968) has been shown to give much better agreement by convolving experimental absorption data with the theoretical Airy’s functions. Alternatively, the excitonic enhancement shown above can be coupled with the Franz-Keldysh

2 OFTICALNONLINEARITIES IN

SEMICONDUCTORS

89

PHOTONENEROY (aV)

FIG. 14. The electric field dependence of the absorption spectrum in bulk GaAs (the Franz-Keldysh Effect) calculated (a) by including Coulombic enhancement and excitonic effects and (b) from Kane band theory without Coulombic enhancement. The curves in (a) are in close agreement with experiment. (After Dohler, 1990.)

analysis to calculate electroabsorption under the assumption that the internal field is small enough to ignore its effect on the exciton. Dohler has indicated the importance of including excitonic effects in calculating the Franz-Keldysh effect (Dohler, 1990). Figure 14 shows a calculation of the Franz-Keldysh result including the full excitonic effects.

b. Modeling Experimental Results Below the band edge, electroabsorption at a fixed wavelength has been shown to be reasonably well modeled by a pure quadratic dependence on field (Partovi and Garmire, 1991). Intimately connected with electroabsorption is electrorefraction,calculated by applying the Kramers-Kronig relation (see below). Calculation of electrorefraction from measurements of electroabsorption gives results that agree well with experiments on photorefractive coupling. Below the band edge, electrorefraction can be modeled to within 20% by a pure quadratic dependence on field, so the Franz-Keldysh effect can be assumed (at wavelengths longer than the band edge) to be described

90

ELSAGARMIRE

by an absorption that depends on electric field E as a z a.

+ a2E2

where typically a2 will depend on the wavelength and the sample purity. There will be a concommitant refractive change:

bn z n2E2 where both a 2 and n 2 are strongly dependent on wavelength. The refractive index change can be determined through the Kramers-Kronig relations, which relate absorption changes and refractive index changes.

3.

KRAMERS-KRONIG RELATION:ELECTROREFRACTION

The Kramers-Kronig (KK) dispersion relation can be derived from first principles by considering the real and imaginary portions of the complex dielectric constant. The result is an integral that relates the absorption and refractive index spectra. If the absorption is known at all photon energies, then the refractive index can be calculated at all wavelengths from the absorption data. In nonlinear optics, the change in refractive index is related to the integral of the spectrum of the change in absorption, as long as all data points are measured at the same excitation level. This enables calculation of the field-induced change in refractive index at photon radian frequency o from values of the change in absorption measured at other photon frequencies w’ as follows (Stern, 1963; 1964; Weiner et a!., 1987):

where E is the static field strength, assumed to be the same throughout the measurement, c is the velocity of light, and P represents Cauchy’s principal value, required in order to keep the integral finite. In principle, the integration must take place over all frequencies 0’. In practice, the electric field-dependent changes in absorption that are considered here occur over only a relatively narrow resonance, so sufficiently accurate values of 6n can be obtained by making measurements only near the band edge (this analysis will not give the linear electrooptic effect). Weiner and colleagues found that calculations based on the K K relation, making these assumptions, agreed with experimental results on Q W modu-

2 OPTICAL NONLINEARITES IN SEMICONDUCTORS

91

lators. The validity of the KK relation in nonlinear optics was investigated in QW structures and shown to be correct by comparing KK calculations of refractive index changes using absorption measurements with measurements of degenerate four-wave mixing. (Chemla et d., 1984). In general, measurements of refractive index change spectra are harder to make, with fewer data points and larger error bars (Koehler et al., 1991). Thus calculating the refractive index change from the Kramers-Kronig relation is an effective alternative to direct measurements.

4. QUANTUM CONFINED STARKEFFECT

Quantum wells have larger electroabsorption and electrorefraction compared to bulk. The first measurements of the electric field dependence of absorption in QWs (Chemla et al., 1983; Miller et al., 1985a) introduced the terminology quantum confined Stark efect. The large magnitude that was observed set the stage for considerable experimental and theoretical work on the electric field dependence of absorption and refractive index in QWs with application to practical modulators. The quantum confined Stark effect (QCSE) is typically an order of magnitude larger than the Franz-Keldysh effect and is important to carrier transport nonlinearities because it is the origin of the optically induced absorption and index changes that are often utilized. a. Physical Understanding

When there are quantum wells, the electric field dependence of the absorption can be calculated from the quantum mechanical solution of particles in a sloping potential well. The effect of an applied field can be understood qualitatively by referring to Fig. 15. The wavefunctions for the electron and hole slightly separate, tending toward opposite sides of the Q W and lowering the oscillator strength; the energy levels also become closer together. Thus the QCSE causes a shift of the resonance to a longer wavelength and a decrease in overall absorption. The electric field dependence of the absorption depends strongly on wavelength; at or above the exciton resonance, the absorption may decrease with applied field because the exciton resonance moves to longer wavelengths and the oscillator strength decreases (Chemla et al., 1983). At operating wavelengths longer than the zero-field exciton resonance, the absorption increases with applied voltage until the exciton resonance wavelength coincides with the operating wavelength, after which the absorption may start to decrease.

92

ELSAGARMWE

FIG. 15. Diagram for understandingthe electric field dependence of absorption in quantum wells (the quantum confined Stark effect, QCSE). Wavefunctions are shown for the quantized electrons in the conduction band and holes in the valence band (a) with no static field and (b) in the presence of a static field E . The quantized resonance energy W,decreases as the field increases. Furthermore, the wavefuncdons move to opposite sides of the well, decreasing their overlap and therefore the oscillator strength.

A t wavelengths sufficiently far below the band edge (with moderate applied fields), the absorption increases quadratically with field, with a coefficient that is an order of magnitude larger than the Franz-Keldysh effect (and the same sign). The refractive index also increases quadratically with field, sufficiently far below the band edge (e.g., 70meV) (Click et al., 1986; Wood ef al., 1985, 1987; Weiner et al., 1987). What makes the QCSE qualitatively different from the Franz-Keldysh effect is the presence of an excitonic absorption resonance, as seen in the zero-field spectrum in Fig. 16 (Chao and Chuang, 1993; Mares and Chuang, 1992; Chuang, 1995). Quantum confinement allows these resonances to be observed at room temperature (in bulk semiconductors, III-V excitonic resonances are seen only at low temperatures). This resonance causes a net decrease in absorption above the exciton as the applied field shifts the excitonic resonance to a longer wavelength. The ability to have either a positive or negative absorption change with applied field, depending on wavelength of operation, is a characteristic that has been used in device design. It is beyond the scope of this chapter to discuss the theoretical derivation of the QCSE, complete with excitonic enhancement. Figure 16 shows experimental results and theoretical analysis that agree in all major features, demonstrating that quantum mechanics can replicate the experimental results of QCSE successfully. Most engineering analyses of QCSE for practical applications use heuristic models that incorporate the QCSE shift, oscillator strength reduction, and excitonic resonance broadening.

2

f.42

OPTICAL

93

NONLINEARITIES IN SEMICONDUCTORS

C46

PHOTON ENERGY (*V)

1420

1440

1460

1480

1500

PHOTON ENERGY (maV)

FIG. 16. Electric field dependence of absorption spectrum (quantum confined Stark effect) in GaAs quantum wells both experimentally (a) and theoretically (b). The field strengths are (i) 0 kV/cm, (ii) 60 kV/cm, (iii) 100 kV/cm, (iv) 150 kV/cm, and (v) 200 kV/cm. (After Chuang, 1995, Figs. 13.7 and 13.8.)

The relation between the QCSE in QWs and the Franz-Keldysh effect in bulk was clarified, at least in the case where excitonic effects could be ignored, by demonstrating that electroabsorption in QWs formally becomes the Franz-Keldysh effect in bulk in the limit of an infinitely thick layer (Miller er al., 1986a). When the potential drop across the layer is small compared with the confinement energy, quantum confinement significantly alters the behavior of the wavefunction. However, with increasing field, numerical analysis demonstrated for a GaAs-like semiconductor that FranzKeldysh-like behavior is recovered once the originally forbidden Q W transitions become strong.

b. Quantum Well Excitonic Resonance Characteristics The reduced dimensionality of excitons in QWs enhances their linear and nonlinear optical properties near the fundamental absorption edge. Measurements of the dependence of the excitonic absorption peak on well

94

ELSAGARMIRE

width (Lee et al., 1988) show that wells of width 5 nm and more have a peak heavy-hole absorption ap that depends inversely on well width L, such that up cc I/Lz. Smaller well widths have peak absorption coefficients that depend less strongly on well width (Jelley et at., 1989), presumably because of increased line-broadening mechanisms. GaAs wells with AI,~,,Ga,,,,As barriers show the following phenomenological result (JeHey et al., 1989):

where the term in brackets is included only for narrow well widths thinner than 5 nm. Otherwise, it is zero. The figure of merit of the QCSE for carrier transport nonlinearities, 6a/a, i s ultimately limited by any residual absorption when the material is in the low-absorption state. Because the highest figures of merit occur for the narrowest lines, understanding residual loss is crucial. One experimental method for making such measurements relates photoluminescence to absorption (Kost et a/., 1989). In p-i-n structures, photoconductivity measurements can be used. Understanding the mechanisms of excitonic line-broadening is crucial to reducing the absorption below the resonance as much as possible. When carriers are totally confined in quantum wells, exciton line-broadening has two components: 1. Inhomogeneous absorption due to interface roughness and alloy

inhomogeneities 2. Thermal broadening, due to homogeneous scattering by LO phonons into bound and continuum states These combine into

r=r,+

rph

exp(hv/kT) - 1

where F0 is determined by inhomogeneous broadening, such as interface roughness and alloy fluctuations, whereas rph is due to homogeneous phonon broadening; hop,, is the energy of the LO phonon. In InGaAs/ InAlAs quantum wells, for example, it was shown (Livescu et af., 1988) that To = 2.3 meV and rph = 15.3 meV for h o p , = 35 meV. This means that even in perfect quantum wells there will always be a substantial broadening of the excitonic resonance at room temperature.

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

95

At least three more mechanisms can contribute to broadening:

3. Finite lifetime for electrons and holes in the wells (due to thermionic emission over the barriers, tunneling through the barriers, or carrier recombination) 4. Free carrier screening of the Coulombic interaction between the electrons and holes, which becomes important at high levels of optical excitation 5. Mixing of states in strained quantum wells

The width of the excitonic resonance has a dependence on Q W width that was predicted theoretically (Lee, 1986) and confirmed experimentally (Jelley et al., 1989) to be constant, r h h = r,, for wider wells (L, > 7.5 nm) and can be modeled for narrower wells by

The enhanced linewidth broadening in narrow wells limits their ability to exhibit large ratios of &//a. The oscillator strength is given by the area of the absorption resonance. Assuming a Gaussian line shape of width Ghh, then the area under the absorption resonance is Ah, = 2.13Ghhahh (Jelley et al., 1989; Masumoto et al., 1985). Since wider wells have a line width that is independent of well width (L, > 5 nm) and /a,,,, varies inversely with well width, it is possible to write the oscillator strength for wider wells as inversely proportional to well width. This was confirmed experimentally (Lee et al., 1986; Jelley et al., 1989) and agrees with calculations of oscillator strength (Masumoto et al., 1985). Narrower wells can be included in the oscillator strength by a phenomenological expression that includes a quadratic term:

For GaAs wells with A1,,,,Ga0,,,As barriers, C, = 2.7C2 (when the well widths are expressed in nanometers). A larger band-gap difference between the well and barrier, which provides more confinement, will give a larger oscillator strength, as has been shown experimentally and is discussed below. It was pointed out that an artificial apparent broadening can result from nonuniform electric fields in MQWs placed in diodes (Little et al., 1989). For this reason, many devices separate the MQW from the doping regions by undoped spacer layers.

96 c.

ELSAGARMIRE

Welt- Width Dependence of the Stark Sh$t

The peak of the exciton resonance moves to longer wavelength quadratically with field. This shift depends to the fourth power on the size of the effective infinitely deep well that determines the wavefunction Leff (Matsuura and Kamizato, 1986; Hiroshima and Lang, 1986):

Note that C , < 0, so that the lowest order transition energy decreases with increasing field. The amount of shift increases as the fourth power of the well-width and quadratically with applied field. Higher order transitions have energies that increase with increasing field. The QCSE occurs from the interplay between the field dependence of the excitonic resonance wavelength and the field dependence of the oscillator strength. The area under the absorption spectrum should remain constant with applied field (Miller et a/., 1986b). This means that the sum of all absorption changes over different parts of the spectrum should be zero. This sum rule has been validated experimentally (Whitehead et al., 1988b). The well-width dependence of QCSE was first measured experimentally (Whitehead et al., 1988a) by observing that the largest contrast ratio comes from long-wavelength operation of thinner wells (a 4.7-nm well had twice the long-wavelength transmission decrease of an 8.7-nm well). However, because wells that are too thin experience broadening of the excitonic resonance, there will be an optimal well width (Jelley et af., 1989). This optimum for maximizing QCSE was determined experimentally by investigating a series of samples in the GaAs/Al,,,,Ga,.,,As material system with different Q W widths. The field dependence of the absorption spectrum was measured in each, and the maximized value of 6a was determined for each. This occurred for a particular wavelength and at a particular field strength. Figure 17 shows the relevant parameters. The field strength and wavelength at which the maximum absorption change was measured are shown as curves 3 and 4, respectively. It can be seen that the required field strength goes up dramatically as the Q W width decreases. The shorter operating wavelength to achieve a maximum index change is a measure of the increased confinement energy within the QW. The value for maximum absorption change at this field strength is shown as curve 1, while the top curve shows the low-field absorption (which is the minimum absorption) necessary to calculate the figure of merit, shown as curve 2. Data such as these are necessary to design optimized devices and a semiempirical model was developed from these data in order to predict device performance (Lengyel et at., 1990).

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

8

6

a' .

z

U

4a

2

0 0.

- 850 E

-m=

t

r(

- 750

FIG.17. Results of an optimization of the quantum confined Stark effect in GaAs/ Al,,,,Ga,,,,,As quantum wells as a function of well width (top curue) the low-field (minimum) absorption; (1) maximum value of absorption change; (2) maximum value of the figure of merit, absorption change divided by minimum absorption; (3) electric field required to achieve maximum absorption change; (4) wavelength at which maximum absorption change takes place. (After Jelley et al., 1989.)

98

ELSA GARMIRE

This analysis of QCSE in single Q W s shows that operating at an optimal electric field and wavelength for maximizing the absorption change per Q W leads to an optimized figure of merit, f = Ga/a-, for each well width. The data in Fig. 17 show that there is an optimal well width that maximizes the figure of merit when the electric field can be varied at will. The maximum f is in 3.5-nm wells, where f = Ga/a- = 5.2 at a wavelength 9nm below the zero-field exciton resonance, where the residual loss a - = 0.43 pm-'. In the analysis of a properly designed reflective Fabry-Perot (FP), I will show (Section IV) that the reflectivity will vary from 0 to R,, = [f/(2 +j)]'.Thus f is an important figure of merit that allows quick analysis of potential performance. For f = 5.2, R,, is predicted to be 52% in the optimized case. It can be seen from the data that 6a cc l/L,: then GaL, is approximately constant. That is, the absorption change per quantum well is approximately constant, independent of well width. For GaAs wells with .41,,,,Ga,,,,As barriers, the maximum absorption change is approximately ciaL, = 1% per Q W , with a residual loss of 6 a - L = 0.15% per QW. impedance matching requires a front mirror reflectivity R, = exp( -2a+L), where a+ is the high-field absorption. For a+ = 1.15% per QW, then an impedance-matched Fabry-Perot (IMFP) containing 50 Q W s will require R, = 32% reflectivity. This is approximately the same as the Fresnel reflectivity of the air/GaAs interface, representing a convenient means for IMFP fabrication. The net result of the well-width studies of Q C S E in electrically driven modulators is that there is very little design flexibility with single Q W s used in normal-incidence modulators in a given material system. The next step is to investigate the dependence of QCSE on well depth, by varying the barrier composition. tl.

Dependence on Well Depth

Another approach to optimization is to vary the composition of the well barriers, investigating the dependence of QCSE on the depth of the potential well. When the incident light wavelength is below the zero-field excitonic resonance, two competing phenomena take place when a field is applied: The absorption tends to rise due to the shift in excitonic resonance, but the absorption tends to fall due to the loss in oscillator strength. The latter effect can be reduced by providing as much confinement as possible, deepening the well by increasing the bandgap of the barriers. The optimization of well width described above used 30% AIAs in the barriers. Increasing the AIAs fraction would give more confinement and larger QCSE. It has been shown that pure AlAs barriers give nearly 50% greater exciton oscillator strength compared with 40% AIAs, particularly at

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NONLINEARITIES IN SEMICONDUCTORS

99

high static electric fields, because of better confinement and more overlap of electron and hole wavefunctions (Pezeshki et at., 1992). The comparison was made in two samples with the same well width, 7.5 nm, varying the barrier height only. This work pointed out that the fact that the barriers have indirect bands (AlAs > 40%) is not optically important because scattering to indirect valleys is slow compared with the exciton ionization time. Impressive photocurrent data imply that the figure of merit at 7.5 V can be as much as f = 20. However, the authors also point out that the advantages of larger absorption changes and simpler fabrication with Al As barriers must be evaluated, for applications as electrically driven modulators, against the possibility of lower saturation intensity and slower photocurrent generation, since carriers can be stuck in deeper wells. Similar considerations must take place in designing MQWs for carrier transport nonlinearities. It must be possible to get the carriers out of the wells if they are to generate the screening fields that are the origin of carrier transport nonlinearities. Modulators that can operate with high intensity without saturation require carriers to be removed easily; that is, they use shallow wells rather than deep wells. GaAs QWs with barriers that contain only a few percent aluminum can exhibit strong room-temperature excitonic features (Goossen et al., 1990a). These shallow QWs have enhanced electroabsorption at small biases because of the ease of ionization. At the excitonic resonance, the transmission of a sample with 2% AlAs in the barriers changed from 29 to 47% with a change in bias from + 1 to -3 V across an i region that contained 50 QWs of width 10nm. Since the total well thickness was 0.5 pm, the resonance absorption changed from a+ = 2.5 pm-' to a- = 1.5 pm-' (Goossen et al., 1991). This resonance absorption is useful in SEED applications, but has too much residual loss for spatial light modulator (SLM) applications. On the long-wavelength side of the excitonic resonance, studies in a reflective geometry show that double-pass loss changed from 5 to 30%. The figure of merit f = Ga/a- = 5, which enables an IMFP reflectivity to change from 0 to 50%. The double-pass loss of 30% means that a front mirror reflectivity of R , = 70% is required to achieve impedance matching, rendering a fairly high finesse FP that will have rather strict tolerances. Easy carrier removal from wells is an important advantage in selfmodulation nonlinearities. There will be an optimal geometry for barriers, based on a tradeoff between carrier confinement to improve QCSE and carrier tunneling for ease in carrier removal. Such a study has not been done, to my knowledge, for Fabry-Perot SLMs, although modeling has been carried out to optimize electroabsorption waveguide modulators (Chin and Chang, 1993) and normal-incidence modulators used in a differential transmission mode (Goossen et al., 1995a), with an analysis of operational and manufacturing tolerances (Goossen et al., 1998).

100

e. Electrorefraction in

EUA GARMIRE

Q Ws

The refractive index change in the presence of QCSE has already been alluded to (Glick et al., 1986; Wood et ab, 1985, 1987; Weiner et al., 1987). Below the excitonic resonance, the dependence on field is quadratic, and the index changes are 6n < 0.03. However, at wavelengths near the peak of the electroabsorption change (9 meV below the zero-field exciton resonance), there also will be an index change that can approach 6n 0.1 on resonance, calculated by applying KK to a calculated absorption coefficient (Hiroshima, 1987). A careful study of electroreflectance and electroabsorption allowed the field dependence of the refractive index to be studied experimentally and compared with theoretical analysis (Kan, 1987). Figure 18(a) shows a theoretical curve of the refractive index change induced by an electric field, determined by KK from the electroabsorption calculation shown in Fig. 18(b). This analysis is for GaAs/AlAs, and comparison with the electroabsorption analysis of Fig. 16 (TE only) shows the improvement in exciton confinement at high field strength due the use of larger-band-gap barriers. It can be seen that there is a wavelength on the high-energy side of the exciton resonance at which the refractive index barely changes with field. While there is also a point on the long wavelength side where the index does not change, this point depends on field strength. These curves confirm the two important features of the index change-it can be large and negative on resonance, but on the long-wavelength side it is positive and much smaller. The first direct measurements of the refractive index change spectra used 150-pm-longwaveguidescontainingtwo GaAs QWs each 9.4 nm thick, within a 3.4-pm-thick waveguide (Zucker el al., 1988; Shimizu et al., 1988). Using the measurements, and the calculated filling factor of 0.0058 per QW, the peak long-wavelength absorption change is calculated to be 6a = 0.75 pm-' for TM and 6ct = 0.6 pm - * for TE polarization. The concomitant index change was both measured and calculated from KK (with excellent agreement)to give a peak T M value of 6n,, = -0.086, whereas the largest TM value on the long-wavelengthside of the resonance was an,, = 0.025.For TE modes, the peak value was Sn,, = -0.065, whereas the largest TE value on the long-wavelength side was dn,, = +0.017. The measured refractive index agreed with the calculated predictions of Hiroshima and showed the existence of at least two wavelengths that had no refractive index change, one on either side of the exciton resonance. The wavelength dependence of both the refractive index change and the absorption change has a large effect on waveguide modulators, as pointed out by Zucker. By applying KK to experimentally measured absorption, the refractive index change spectra were analyzed as a function of voltage (Boyd and

-

+

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

101

s

J -LU'

830

840

850 Wavelength

Wavelength

860

1

870 0

(nm )

(nm )

FIG. 18. (a) Electrorefraction spectra (percent refractive index change) in 10-nm GaAs/ AlAs quantum wells based on Kramers-Kronig calculation from calculated absorption spectrum (b) for field strengths from 0 to (i) 50kV/cm, (ii) 80kV/cm, (iii) 120kV/cm, (iv) 15OkV/cm, and (v) 170kV/cm. Maximum index change below the band edge is 6n = +0.01, while on resonance the index change is -0.07. (After Kan, 1987.)

Livescu, 1992). These data confirmed the wavelength (on the high-energy side of the exciton resonance) for which the index change is essentially zero, independent of applied field. However, on the long-wavelength side, the null in index change depends on applied field. Depending on device design and intended application, this index change (typically less than 0.02) may

102

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improve or hinder performance, but it certainly must be taken into account. A theoretical model that allows both the electroabsorption and electrorefraction to be calculated has been shown to agree with experimental FP results (Lin er al., 1994a; Kan et a/., 1987). j : Polurization Dependence

There is a strong dependence on the polarization of light with respect to the plane of the QW. Theoretical analysis (see Chuang, 1995, p.564) indicates that the absorption is proportional to the matrix element squared, multiplied by the following polarization-dependent factor: Polarized in the QW plane: Polarized normal to the QW plane:

Heavy hole Light hole Heavy hole Light hole

312 1/2 0 2

The full import of these factors is not seen in experiments because the excitonic absorption rides on a continuum absorption. Nonetheless, measurements and theory, as shown in Fig. 16, support this polarization dependence. Measurements in devices operated with light at normal incidence to the plane of the QWs do not exhibit these polarization effects, since all light is polarized in the Q W plane. Waveguides are necessary to observe the polarization dependence of the QCSE (Weiner er a/., 1985; Dutta and Olsson, 1987). Considerable research has been undertaken to optimize waveguide modulators based on QCSE, but this is beyond the scope of this chapter, and readers are directed to the research literature (e.g., Wood, 1988; Koch, 1991).

5. ADVANCEDQUANTUM CONFINED STARKCONCEPTS

This section outlines attempts to improve electroabsorption figures of merit over those available from QCSE in isolated single quantum wells (SQWs). First, I investigate what happens if more than one QW are coupled together, considering double-coupled quantum wells (CQWs), in both symmetric and asymmetric geometries. An important distinguishing feature of CQWs is that the long-wavelength absorption decreases with increasing field, which is opposite to SQWs. The tradeoffs are that using CQWs can enable larger 6a values, but usually at the cost of increased residual

2 OPTICAL NONLINEARITES IN SEMICONDUCTORS

103

absorption a- so that the figure of merit is not necessarily larger. Second, superlattices of coupled QWs operating by means of Wannier Stark localization (WSL) will be described. These have the advantage of not using excitonic resonances and are particularly suitable for long-wavelength waveguide applications. Third is an approach that uses strain to move the relative positions of the heavy- and light-hole resonances. Anisotropic strain provides normal-incidence modulators with an opportunity for polarization anisotropy that can provide high-contrast modulation. In GaAs QCSE, results to date indicate that single QWs have the best long-wavelength figures of merit. We will see, however, that this is not necessarily true in other materials, particularly those of interest for longerwavelength devices. This section describes only a few of the many results reported to date; advances are being made rapidly, and readers are advised to watch for further developments.

a. Coupled Quantum Wells In an effort to observe larger absorption and refractive index changes, a number of researchers have investigated coupled QWs (CQWs) in an applied field: two symmetric wells, two asymmetric wells, graded gap, three or more wells. Symmetric wells. The first reports replaced a single QW (SQW) by two coupled symmetric QWs of roughly half the full width. For light in a waveguide polarized normal to the plane of the CQW (TM), the electroabsorption was observed to be comparable with that of an SQW (Islam et al., 1987); this polarization experiences light-hole absorption only. Because of line broadening at high fields, there was no obvious improvement to the contrast ratio in these CQWs. In the TE polarization (which is the only polarization available to normal-incidence light), the symmetric CQWs were shown to have a total absorption at the light-hole resonance larger than at the heavy-hole resonance, a relation opposite to that observed in the SQW. This was attributed to the tighter confinement in the half-size wells. With increasing field, this absorption became weaker, but the next heavy-hole resonance became allowed, increasing the absorption. A mixing of light- and heavy-hole transitions increased the band-tail absorption at high field above that observed for SQWs, limiting the usefulness of these symmetric CQWs. The ability of a static electric field to decouple the quantized levels in symmetric CQWs (5.4-nm GaAs wells separated by 0.6-nm AlGaAs barriers) and thereby shift the absorption edge to the blue was demonstrated at 77 K (Onose et al., 1989). Room-temperature observation of the blue shift was

104

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seen in symmetric CQWs (4.8-nm GaAs wells separated by 3.6-nm AlGaAs barriers) by observing optical nonlinearities in nipi structures (Kost et al., 1989). The magnitude of the observed absorption change was comparable with that measured in SQWs, but the sign was opposite below the exciton resonance; decreasing field increased the long-wavelength absorption. A study of strongly coupled symmetric CQWs of varying dimensions theoretically predicted that 6a = 1.6 p n -' should be possible on the short-wavelength side of the exitonic resonance (using a 0.57-nm AlAs barrier separating 4.2-nm GaAs wells) as a result of field-induceddecoupling (Chan and Tada, 1991). However, experimental results showed considerable excitonic broadening, with absorption changes near the excitonic resonance riding on a large zero-field absorption. The resulting performance was typically not better than that of an optimized SQW. Figure 19 shows some experimental results in GaAs/AlAs CQWs and illucidates how the electroabsorption features depend on field strength. Figure 19(a) shows more moderately coupled wells, separated by five monolayers of AIAs. The application of a field restores the excitonic resonance by reducing the interwell coupling while moving the excitonic resonance only very slightly to shorter wavelengths. However, there is an increasingly large long-wavelength absorption tail due to residual coupling, which limits the ability of the CQW to reduce long-wavelength absorption with increasing field. This becomes clearer in Fig. 19(b), where the wells are separated by only two monolayers of AlAs barrier. The spectrum is rich with absorption features, but this very richness makes it difficult to find a wavelength at which there can be a large figure of merit for modulator applications.

+

Asymmetric coupled quantum wells. Several measurements were made with asymmetric coupled quantum wells (ACQWs) to search for improved contrast ratio. The first room-temperature electroabsorption measurements at normal incidence (Little et al., 1987) gave results that were complicated by the presence of photoinduced field screening, the carrier transport nonlinearity. Nonetheless, an analysis of their data indicates that the absorbance changed from 0.38 to 0.30 as the applied field of 115 kV/cm was screened. The structure consisted of 8- and 4-nm GaAs wells separated by a 5-nm Al,,,Ga,,,As barrier. This is an absorption change per double well of 0.2%, small compared with an optimized SQW. Low-temperature measurements allowed for careful analyses of the transitions possible between the various levels in ACQWs and showed that considerable flexibility in design is possible (see, for example, Le et al., 1987; Golub et a/., 1988; Tokuda et al., 1989). However, room-temperature performance near the excitonic resonance, such as required for normalincidence modulators, does not show improvement over optimized SQWs.

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

700 Wavelength [nm]

720

740 760 780 Wavelength [nm]

105

800

FIG. 19. Photocurrentspectra of coupled quantum wells consisting of two GaAs wells, each 15 monolayers (4.2 nm), separated by (a) 5 monolayers of AlAs (1.42nm) and (b) 2 monolayers of AlAs (0.57 nm).(After Chan and Tada, 1991.)

Waveguide measurements (Steijn et al., 1989) on ACQWs with 8.5- and 4.3-nm GaAs wells separated by a 2.1-nm AlGaAs barrier showed the same order of magnitude absorption change as the uncoupled wide well, with a rough quadratic dependence of absorption and refractive index on internal field at wavelengths below the band edge. However, there were particular resonant voltages at which the absorption could be as much as 30% higher than for a comparable SQW, and the absorption and index changes were shown to be nonmonotonic functions of internal field due to the QW coupling resonances. Such enhancements occur well below the band edge and are useful only in waveguides. An interesting study of CQWs showed that with careful design it is possible to obtain a reasonable contrast ratio in an electroabsorption FP modulator while also having zero chirp (Trezza et al., 1997). Chirp occurs when rapid switching involves a refractive index change as well as electroabsorption. Chirp adds a frequency broadening to a switched optical signal, due to the time dependence of the refractive index. In order to avoid this chirp, it is necessary to find a wavelength of operation where there is an

106

ELSAGARMIRE

electroabsorptive change only and the electrorefractive change is zero. This is possible on the short-wavelength side of the excitonic resonance, as pointed out in several examples above. Using an asymmetric structure consisting of a 5-nm GaAs well coupled with a 2-nm In,,,Ga,,,As well through a 1-nm A1,~,,Ga0~,,As barrier, the absorption changed from x + = 2.2pm-’ to a- = 0.35pm-’, with zero change in refractive index at an applied electric field of 90 kV/cm. This gives a figure of merit f = 6a/ x - = 5.3, or B = 0.84, high enough that in an IMFP, the “on-state’’ reflectivity can be as high as 54%. The devices reported as modulators used 10 QWs, and modeling predicts that a front mirror of reflectivity R , = 0.7 would provide impedance match, since here the CQW L, = 8 nm. In addition to two coupled wells, investigators have analyzed theoretically three or more coupled wells in order to optimize field-dependent effects. For example, by using a five-step asymmetric CQW with modified potential, a large field-induced refractive index change is proposed that will not have a red shift of the absorption edge (Feng et a/., 1997). While the peak index change is not larger than for a single QW, below the band edge (where the devices are most practical), the refractive index change is an order of magnitude larger than for an SQW and the absorption change is smaller. Further studies of coupled quantum wells have been carried out for longwavelength applications. These will be discussed later. Wells with graded composition profile. Applying fields to graded composition wells, rather than to step-index wells, has an effect comparable with that of coupled asymmetric wells. The first suggestions for graded-gap QWs were made in 1987 (Nishi and Hiroshima, 1987; Hiroshima and Nishi, 1987; Sanders and Bajaj, 1987). Effective gradation of composition can be achieved by disordering through interdiffusion of constituent atoms across the well-barrier interfaces. Calculations show that for the same voltage, a 30% increase in electroabsorption change is expected (Micallef et al., 1995). Apparently no optimization has been carried out and no experimental results have been reported, however. A stepped-well structure can have many of the properties of graded-gap wells. The use of two materials within the well that have a type I1 lineup relative to each other but are type I relative to the barrier material can provide spatial separation of the electron-hole pair in the ground state at zero applied field. Such a structure is predicted to provide a large blue shift of the absorption edge on application of an electric field (Stavrinou et al., 1994; Suzaki et a/., 1991). An equivalent structure that has been modeled uses delta doping on either side of a QW to provide an internal bias to each QW (Batty and Allsopp, 1995).These are examples of some of the complex structures that are being analyzed theoretically for possible improvements to QCSE as modulators.

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

b.

107

Wannier Stark Localization

In the limit of a large number of coupled quantum wells, a new phenomenon comes into play, called Wannier Stark localization (WSL). When there is no field applied to an array of coupled quantum wells, the quantized electron and hole levels form a miniband. However, a static field can cause the states to localize (Bleuse et al., 1988a; Voisin et al., 1988; Mendez et al., 1988), which will introduce a strong blue shift to the absorption. Since the phenomenon does not depend on any excitonic resonances, WSL can be particularly valuable for long-wavelength operation in narrow-gap semiconductors, which do not show particularly large excitonic resonances (Bleuse et al., 1988b). As in CQWs, the sign of the absorption change will be opposite that of QCSE. In the first room-temperature measurements (Bar-Joseph et al., 1989), an applied field created absorption changes (averaged over the entire QW region) of ( 6 a ) = 0.77 pm-' due to WSL in 100 periods of 3-nm GaAs wells and 3-nm Al,.,Ga,,,As barriers. The absorption change was 0.46% per well. While less than the value for an optimized SQW in GaAs, large modulation could be obtained over a substantial spectral range. This demonstrates that WSL can provide practical modulators for telecommunication applications where a wide wavelength range is desirable. The authors observed localization of the heavy holes at 2 kV/cm, whereas the electrons were not localized until 80 kV/cm and even then were not fully localized, resulting in some residual absorption. The WSL in 3-nm GaAs wells and 3-nm AlGaAs barriers was used in an asymmetric Fabry-Perot reflection modulator (Law et al., 1990b); the absorption changed from 0.78 to 0.28 pm- (averaged over the wells and barriers) when the voltage changed from 0 to -8 V. The large residual absorption was caused by indirect transitions, even at high fields. A comparable value was measured in transmission (no FP) using 110 periods of 3.5-nm barriers and 3-nm wells (Olbright et al., 1991); 6a = 0.42pm-' was reported. These results indicate that if there is a strong excitonic resonance, optimized SQWs are more likely to give a comparable absorption change per Q W but have a smaller residual absorption and be more practical for normal-incidence modulators. Measurements of GaAs/AIGaAs WSL in waveguides reported that very low drive voltage with high extinction is possible well below the exciton resonance (Moretti et al., 1992). The waveguide configuration is particularly applicable to WSL at long wavelengths, such as 1.55 pm, where excitonic features are weak. Since I am confining this chapter to surface-normal devices, I will evaluate waveguide results only when they can predict surface-normal performance.

108 c.

ELSAGARMIRE

Strained Quantum Wells

The introduction of strain into quantum wells can alter their response to electric fields. A number of investigations have been made of strained QWs aiming to improve the QCSE. The first characteristic of strain is that it removes the degeneracy between the light and heavy holes at the center of the Brillouin zone. This means that either light holes (LHs) or heavy holes (HHs) have minimum energy, depending on whether the strain is compressive or tensile, respectively. However, quantum confinement also lifts the light/heavy-hole degeneracy. Thus, depending on design, strain can add to or subtract from the LH-HH splitting observed in the absorption spectra of unstrained QWs (Hong et al., 1988; Kothiyal et al., 1987). The key requirement for normal-incidence electroabsorption modulators is a sharp excitonic transition. This requires low operating temperature, good interface quality control, good well-to-well size control, and uniform electric field. Once these have been achieved, improvements are possible, in principle, by using strained QWs to increase the absorption coefficient. If substantial improvment can be demonstrated, then the challenge is to grow a thick active region without strain buildup causing dislocations. When epitaxial layers with different intrinsic lattice constants are grown on top of each other, each layer tries to take on the lattice constant of the substrate. This is what produces the strain that splits the LHs and HHs. To predict operation of strained-layer material, a few facts can be used: 1. The strain is biaxial, occurring in both in-plane directions; the layer is not strained in the out-of-plane direction. 2. Layers with the same lattice constant as the substrate will be unstrained, H H s and LHs will be degenerate in the absence of quantum confinement. 3. Layers with a larger lattice constant than the substrate will be compressively strained, with HHs having a lower energy than LHs, in the absence of quantum confinement. 4. Layers with a smaller lattice constant than the substrate will be tensilely strained, with LHs having a lower energy than HHs, in the absence of quantum confinement. 5. If the substrate is removed, the strain is shared between the well and barrier layers.

In the presence of biaxial compressive strain, HHs have lower energy than the LHs, whereas tensile strain (in the absence of quantum confinement) causes LHs to have lower energy than HHs.

2 OPTICAL NONLINEARITES IN SEMICONDUCTORS

109

Compressive strain. Quantum confinement causes splitting of HHs from LHs, with the HHs having lower energy. Biaxial compression of the wells further splits HH and LH resonances, with HHs having lower energy. In some cases of compressively strained wells, LHs will no longer be confined (InGaAs wells, GaAs barriers and substrate, operating at 980 nm) (Van Eck et al., 1986) or will be only marginally confined (InAs/GaAs strained-layer short-period superlattice wells, GaAs barriers and substrate, operating at 980nm) (Hasenberg et al., 1991). Further results on InGaAs/GaAs will be reported below. Tensile strain. Tensile strain causes LHs to have a lower energy than HHs. This counteracts the effect of quantum confinement, so it is possible for the H H and LH subbands to merge, to give particularly large absorption. (Hong et al., 1988). The splitting between LH and HH valence subbands can be eliminated through careful manipulation of strain and quantum-size effects. The normal ordering of the uppermost subbands can even be reversed so that the LH valence subband lies at the highest energy. Furthermore, in waveguides, the TE mode couples to HHs, while the TM mode couples to LHs, so the HH-LH splitting is expected to play an important role in determining the electrooptical properties of QWs in waveguides. The merging of LH and HH subbands was demonstrated at 77K by observing the spectrum as a function of applied field (Gomatam et al., 1993a). A room-temperature comparison of the modulation depth in comparable tensile-strained and unstrained QWs demonstrated increased modulation depths at low drive voltages in tensile-strained devices, although the difference was less at the highest voltages (Gomatam et al., 1993b). Normal incidence measurements were made on 10 strained QWs of thickness 9.5 nm and 11 unstrained 4.6-nm QWs. The smaller unstrained QW was chosen to have its excitonic resonance at the same wavelength as the strained well at the field required to achieve merging in the tensile-strained case. Considerable sample-to-sample variation was noted and attributed to crystal growth difficulties. Nonetheless, by measuring the two-pass tensile-strained reflection (off a Bragg mirror), it was found that at -6V the long-wavelength absorption changed by 2.7 times more in tensile-strained QWs than in unstrained QWs, although for higher fields (- 15 V and beyond), the absorption changes were comparable. It was shown that tensile strain could remove the polarization dependence of electroabsorption (Chelles et al., 1994; Wan et al., 1994) and electrorefraction (Zucker et al., 1992) in waveguide modulators operating at 1.55 pm. The reported electroabsorption figure of merit was 6a/a- = 3.3, with a low-field absorption coefficient of a- = 0.024pm-'. This low absorption is

110

ELSAGARMRE

useful for waveguides only. Further discussion of waveguide devices is beyond the scope of this chapter, except to point out that proper design of tensile-strained Q W structures can provide potentially chirp-free characteristics (Yamanaka et al., 1996). Anisotropic strain. Anisotropic strain can be used to create different QCSE for two polarization directions at normal incidence. Polarization rotation will result, which can be analyzed through crossed polarizers, creating a high-contrast modulator. Anisotropic in-plane strain has been created by bonding GaAs/AlGaAs MQWs at high temperature to a lithium tantalate substrate and utilizing differential thermal expansion (Shen et al., 1993a, 1993b). This principle was used, under conditions of unaxial strain, to demonstrate a highcontrast transmissive optically addressed modulator (Shen et al., 1994). This p-i-n modulator is based on the principle that the strong anisotropy in absorption and refractive index decreases with increased field. When the incident light is polarized linearly at 45 degrees with respect to the strain axis, the birefringence induced by the in-plane anisotropic strain causes the transmitted light to be elliptically polarized, mainly due to the anisotropy in absorption. A crossed polarizer will still transmit some of the light; this is the “on state” of the modulator. Application of a suitably high static field (14V) can remove this anisotropy so that the analyzer now will have null transmission, the “off state.” In a device that contained 150 QWs 8 nm thick grown in a p-i-n structure, the “on state” insertion loss due to the crossed polarizer was 7dB (20% transmission), while the measured contrast ratio was 5000:1, depending on the quality of the polarization analyzer. A polarization rotation as large as 20 degrees was measured for fields of 6.7 V/pm. The maximum absorption anisotropy observed was 1 pm-’ at the heavy-hole resonance, which decreased rapidly with voltage. The refractive index anisotropy was 0.03 at the heavy-hole resonance and was essentially independent of voltage. The absorption anisotropy decreased by an amount 0.54pm-’ at the heavy-hole exciton peak for a bias of 14V. Studies were made of anisotropic biaxial strain, with one direction in compression and the other in tension (Huang et al., 1995). Devices were constructed using the same differential thermal expansion technique just described. Both compression and tension cause the excitonic resonance to move to a longer wavelength. When the light was polarized along the compressive direction, an increase was measured in the heavy-hole absorption coefficient, along with a decrease in the light-hole absorption. The opposite occurred for the tensile-strained polarization. The modulator contained 30 GaAs wells 7.5 nm thick with 10-nm AlGaAs barriers. A peak Q W absorption change of Sa = 1.4pm-’ was measured on the short

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

111

wavelength side at a field of 7.4 V/pm using light polarized in the direction of compressive strain. The absorption change was 50% higher than in comparable unstrained material. The short-wavelength figure of merit was Ga/u1. The long-wavelength figure of merit was not as good as in comparable unstrained QW, however, because of excess excitonic line broadening; a maximum of 6a = 0.93 pm- was measured with 6a0.7 pm- Light polarized in the tensile-strained direction experienced much smaller QCSE. The absorption anisotropy is much larger in the unaxial strain case, which is the most effective way to achieve a high-contrast transmission modulator using polarization rotation.

-

'.

'

-

Internaljields with [111 J growth. Growth on [ l l l ] planes creates large internal electric fields generated by the piezoelectric effect (Smith and Mailhiot, 1987; Mailhot and Smith, 1988). These fields lie in the direction of crystal growth and thus provide a sawtooth band structure (Laurich et al., 1989) that can be strongly affected by external fields. Measurements at 77 K showed that in the absence of an external field, no quantum confinement was observed in absorption (Caridi et d., 1990). The zero-bias photocurrent spectrum beyond the bulk GaAs absorption band edge was essentially featureless. This was attributed to localization of electron and hole wavefunctions to opposite sides of the QW, leading to a loss of oscillator strength in the exciton absorption. Without external bias, the internal field was calculated to be 140 kV/cm. A reverse bias external field, applied through a p-i-n structure, reduced the field to 80 kV/cm, and the characteristic quantum confined absorption spectrum was retrieved. From measurements of the shift in the exciton position with applied reverse bias, the authors demonstrated that the intrinsic strain-generated built-in electric field was 170 kV/cm, in agreement with the predictions. The built-in field can be reduced by an external bias to obtain a blue shift of the absorption edge. The magnitude of the required field depends on the total lengths of the wells and barriers in the MQW structures (Pabla et al., 1993). When MQWs are incorporated in the i region of length L within a p-i-n diode, the field in the wells E , can be written as

where V is the sum of the applied voltage and the built-in voltage drop within the p-i-n diode, E,, is the piezoelectric field, and L, is the total width of the barrier material. Samples of the same i length L and same individual QW length L, but with different numbers of QWs, M y can be compared by noting that Lb = L - ML,. The field in the Q W will become zero when V / L = E,(1 - MLJL). At zero applied voltage there is typically a large

contribution in the well from the piezoelectric field. Increasing the voltage decreases the field across the wells, resulting in a QCSE blue shift. Eventually, the applied field may cancel the piezoelectric field, maximizing the QCSE shift and &/a. Figure 20 shows an example of the photocurrent spectra for piezoelectric InGaAs QWs 10 nm thick grown on (1 ll)B GaAs in a geometry incorporating 17 QWs in an i layer 0.62 pm thick. The strong piezoelectric field at zero applied field produces a QCSE high-field spectrum that has a strongly red-shifted exciton. As the p-i-n diode is biased, the applied field tends to cancel this piezoelectric field, which can be seen by the recovery of the excitonic resonance at - 4 V. Further increases in voltage, however, broaden out the excitonic resonance, reducing the oscillator strength, presumably because of enhanced tunneling due to high barrier fields. The use of piezoelectric fields to prebias Q W s makes low-voltage modulators possible. The preceding analysis means that as the number of QWs is increased, the voltage required to reach zero internal field goes down. A contrast ratio of 4.51 was demonstrated with only 3 V applied when 25 wells were placed within the 0.62-pm i region. A low-temperature (10 K)excitonic optical nonlinearity has been demonstrated in piezoelectric Q W s grown on [lll] substrates that uses photoinduced screening of this internal field (Sela et al., 1991). Picosecond

Y

920

940

960

9ao

Wavelength (nm.) FIG. 20. Photocurrent spectra of InGaAs QWs grown on (1 I l)B GaAs in the presence of internal piezoelectric fields. At a bias of approximately - 5 V, the spectrum regains its excitonic resonance by canceling the internal field. Too high a field ( 2- 8 V) causes broadening and loss of oscillator strength. (After Pabla ef a/.. 1993.)

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measurements of carrier dynamics at 77 K have separated the contribution of in-well screening from that of long-range screening associated with carriers that have escaped the wells (Harken et al., 1995).

Strained coupled quantum wells. Incorporating both strain and coupled quantum wells provides considerable design flexibility. This has been used to demonstrate a blue Stark shift in InGaAs/InP QWs, both theoretically and experimentally (Gershoni et al., 1990). Calculations have been made of a macroscopic piezoelectric effect, based on strain-induced changes in the confinement of carriers in an asymmetric QW structure (Khurgin et al., 1989). The calculations show a piezoelectric coefficient estimated to be a few percent of that in quartz. The removal of the center of inversion symmetry by the asymmetric Q W is the mechanism.

6. QCSE AT OTHERWAVEGUIDES Most of the work on QCSE reported in preceding sections was for GaAs wells and AlGaAs barriers, producing modulators that operate at wavelengths from 750 to 850 nm. Considerable research has been performed at other important wavelengths, both for surface-normal applications and for waveguide applications. Excitonic features are strong in wide-band-gap materials, and these have been demonstrated to possess large QCSE. At longer wavelengths, the excitonic resonances become weaker, and the more complicated structures have been studied in an effort to increase the electroabsorption and electrorefractive figures of merit. Here is a compilation of some of the more interesting results.

a.

Visible Wavelengths

Visible wavelengths require wide-band semiconductors. Electroabsorption was demonstrated near 610nm (Partovi et al., 1991) using 50 11-VI strained quantum wells 6.7 nm wide consisting of CdZnTe with 7.7 nm ZnTe barriers grown on a CdZnTe buffer layer on top of GaAs. The maximum long-wavelength absorption change was 6a = 1 pm- at 138 kV/ cm (the absorption coefficient averaged over the MQW region was ( d a ) = 0.46pm-') with a residual absorption of a0.43pm-'. The maximum QW figure of merit was f = ba/a- = 2.3, predicting an impedance-matched Fabry-Perot (IMFP) reflectivity in the on-state of 30% with a front mirror reflectivity to 40%. Devices require ion implantation to reduce dark conductivity and thermal heating at high fields.

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ELSA GARMIRE

An alternative for visible operation is based on the GaAs/AlGaAs system, using high aluminum concentration in the wells and AlAs barriers (Goossen et al., 1991). Even though the wells may be indirect (for > 40% AlAs), the principle is that since the absorption due to momentum or spatially indirect transitions is so weak, these transitions may be practically ignored when designing electroabsorption modulators. In an indirect material, a large, sharp absorption step exists at the direct band edge, which may be modulated. In addition, strong roomtemperature excitons at the direct band gap are observed in these indirect quantum-confined systems. Using AlAs barriers and wells containing 30% AIAs, a long-wavelengthdifferential transmission change of 20% corresponding to an absorption change of 6a = 0.5 pm- was measured for - 10 V applied, with a figure of merit Sa/a- z 5, comparable to an optimized GaAs/AlGaAs modulator near 850 nm. In an IMFP this should make a high-contrast modulator with 53% reflectivity in the on state, using a front mirror reflectivity of 62%. Contemporaneously, preliminary results on the modulation of a 633 nm HeNe laser were reported (Pezeshki et al., 1991), using reflection through similar Q W s (A1,~,Ga0~,Aswells and Al As barriers). The fractional reflectivity changed by 7% (from 24 to 26%), and normalized photocurrent measurements indicated a promising figure of merit. Yet another approach to the visible wavelengths is wide-band 111-V QWs (Blum et al., 1994). Using 25 QWs of 10-nm In,,,,Ga,,,,P wells and 10-nm l n o , 4 9 ( A l ~ , ~ G a o ~P5 )barriers, ~,5, measurements showed 10% change in transmission on the long-wavelength side of the exciton for - 4 V applied. At this wavelength the figure of merit Scl/a- from photocurrent data appears to be about 5, potentially useful in a high-contrast IMFP modulator.

'

b.

980nm

A n important wavelength for surface-normal applications is 980 nm, the wavelength of strained InGaAs grown on GaAs substrates. The importance of this technology for SLM applications is that the substrate is transparent to the InGaAs exciton wavelength. As a result, the substrate does not need to be removed in order to perform transmission measurements. Experimental applications such as image processing would use strained-layer lasers, operating at 980 nm, as sources (which were developed for pumping erbiumdoped fiber amplifiers). More recently, it has become clear that optimized devices typically will be used in reflection, incorporating mirrors grown between the modulator and the substrate, which removes the advantage of a transparent substrate. Measurements of QCSE in strained InGaAs Q W s were first made using 10-nm wells (Van Eck et a!., 1986). Because InGaAs has a larger lattice

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constant than the GaAs barriers and substrate, these QWs are compressively strained. Nonetheless, good QCSE characteristics were reported. The figure of merit appears to be optimized at 3V (54kV/cm) and at 23meV below the zero-field exciton resonance, with a maximum absorption change (QW only) of 6a = 0.35 pm-' and a residual zero-field absorption of a - = 0.075 pm- yielding &/a- = 4.6. This figure of merit is only slightly less than in GaAs. However, since the optimal operating wavelength is farther from the exciton resonance, the absorption change per QW is low enough that a normal-incidence IMFP would require high-reflectivity mirrors and therefore be sensitive to temperature and to fabrication tolerances. Ternary wells have alloy scattering and wider exciton resonances than binary wells. It is also difficult to grow a large number of wells (desirable to lower the required mirror reflectivities) before strain builds up and reduces the quality of the crystal growth. There is motivation, therefore, to use a binary short-period strained-layer superlattice (SPSLS) in the QW region of modulators (Jupina et al., 1992). The SPSLS consists of a few monolayers of InAs alternating with a few monolayers of GaAs, until the thickness of a QW is reached. This achieves effective ternary material without alloy scattering. The measured QCSE in the SPSLS is comparable with that observed in ternary QWs. Because optimal figures of merit occur on the long-wavelength side of the excitonic resonance, it is important to characterize the long-wavelength tail (Kost et al., 1988). I will define a wavelength shift ,It from the excitonic resonance peak to where the absorption has dropped to 10%. The strained ternary QW (Van Eck et al., 1986) gave 2, = 10nm (12meV), whereas the initial SPSLS results (Jupina et al., 1992) gave ,It = 20 nm. However, later growths (Huang et al., 1997) showed that 50 wells can be grown with ,I,= 12 nm, only slightly larger than the ternary 10-well result. At 14 nm from the exciton resonance, with a zero-field residual absorption of 750cm-', the figure of merit is &/a- = 4.7, comparable with the Van Eck case, but with the ability to grow more QWs. The reflective FP design model indicates that the 50 QWs, can impedance match and achieve an FP reflectivity variable from 0 to 50%, using a front mirror reflectivity of 66%, offering real promise as a high-contrast, low-loss SLM. Electrorefraction measurements in SPSLS at wavelengths near the optimal electroabsorption, determined both by direct interferometric measurements and by calculation using the KK relation (Koehler et al., 1991), showed 6 n < 0.008, and very sensitive to wavelength (at a given applied field), just as in the GaAs/ AlGaAs case. An asymmetric Fabry-Perot structure was grown in the InGaAs/GaAs system using 50 QWs and Bragg reflectors. This device showed a change in

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reflectivity from 30% to 0 with the application of - 16 V (Hu et al., 1991). Spectral measurements showed that between 0 and 8 V the change in reflectivity was almost entirely electroabsorptive in nature, but electrorefractive contributions began to be important at higher internal fields. The maximum reflectivity change that could be achieved was limited by the small band-gap difference between InGaAs and GaAs, particularly at low indium levels, resulting in weak excitonic features that could easily be ionized in an electric field. Higher confinement of the electron and hole is necessary to achieve higher reflectivity changes, and this can be achieved by adding aluminum to the barriers. AlGaAs barriers increase the confinement of the carriers in the wells and greatly improve their QCSE properties (even though the low-temperature luminescence is degraded because of high aluminum concentration grown by MBE at a rather low temperature) (Pezeshki et al., 1991). Absorption spectra show figures of merit increasing with increasing aluminum concentration in the barriers. With A1,,,,Ga0,,,As barriers and 7.5 nm Ino,,Gao,,As wells, the absorption at wavelength 1.016 pm changed from ct- = 0.033pm-' to a+ =0.63pm-' at 15V, for a figure of merit of 6a/(x- = 19, or /I= 0.95. These experimental values, when inserted into the scaling laws derived in Section IV, predict that an IMFP reflection change as high as 83% is possible. Indeed, Pezeshki et al. (1991) demonstrated a 30-well IMFP that was almost impedance-matched (2% reflectivity) in its high-loss state with 21.5 V applied. The reflectivity increased to 77% at 0 V, for a 37/1 contrast ratio. The scaling laws predict an optimum front mirror reflectivity of 75%; the experimental IMFP used a 5-period GaAs/AlAs B r a g reflector with a predicted reflectivity z 80%. One disadvantage of this structure, it was pointed out, was that the high barrier to carrier escape may result in a low saturation intensity, since the carriers have more difficulty tunneling out of the wells and being collected in the doped regions. Similar structures also have been grown by OMVPE (Buydens et al., 1991, 1992), although this paper reported a maximum reflection change of only 27% for 7 V applied in an FP that included 50 QWs 10 nm wide. One reason for this significant difference is that Pezeshki's F P was well impedence matched; the other was not. Clearly, these strained InGaAs materials show the potential for excellent modulators, but the challenge is to achieve any desired wavelength. Further optimization was pursued by using AlAs barriers (which have indirect band gaps), and seeking to push strained InGaAs as far as possible into the infrared. Examination of the room-temperature QCSE in such structures (Ghisoni et al., 1994) shows the characteristic Stark shift with applied field and retention of excitonic strength up to very large applied electric fields, comparable with that in unstrained GaAs/AlGaAs MQWs. In particular,

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clearly resolvable excitons were retained for fields up to -300kV/cm. Photocurrent measurements of Ino. +3a,.,,As QW lO-nm wide indicate a figure of merit Sa/a- x 10 at long-wavelengths out to 960nm, but falling rapidly at longer wavelengths.

c.

I.06jun

One particular wavelength of great interest is 1.06 pm, the wavelength of operation of the NdYAG laser. The first measurement of QW modulation at this wavelength was in waveguides, where an orientation-dependent phase modulation was observed (Das et al., 1988). To achieve normalincidence modulation, an attempt was made to use the strained ternary InGaAslGaAs QW and to increase the indium content, thereby pushing the operative wavelength out to 1.06 pm. The QCSE was measured for several samples fabricated in this way (Woodward et al., 1990); it was found very important to introduce a strain-relief layer between the QW and the GaAs substrate. While these samples had considerable variability due to strain nonuniformities, the figure of merit was as high as Sa/a- = 4.3 at 1.064 pm wavelength with 9V applied across sample B (50 QWs grown on a strain-relieving buffer). These numbers, along with the maximum QW absorption a+ = 0.2pm-', are high enough for possible use as IMFP modulators. Sharper resonance features, with much smaller long wavelength parameters, were reported using InP substrates with 50 InAsP strained ternary wells 10 nm wide grown by chemical beam epitaxy (Woodward et al., 1991). A fractional transmission change of 15% was reported at 1.064 pm with 10 V applied. The figure of merit appeared to be Sa/a- = 3, with a maximum QW absorption of a+ = 0.4pm at 10 V and a long-wavelength parameter of 12 nm, better than when using GaAs substrates. Inserted into an IMFP with a front mirror reflectivity of 67% would provide a maximum reflectivity of 36%. Further work on modulators for the 1.06-pm wavelength range investigated lattice-matched quaternary QWs and strain-balanced ternary wells and barriers grown on InP substrates. Considerable limitation resulted from alloy disorder as well as nonuniform QW composition and thickness, particularly due to temperature variations during epitaxial growth (Woodward et af., 1992). The best results were reported for strain-balanced material grown on InP with compressively strained InAsP wells and tensilely strained InGaP barriers. Here the long-wavelength parameter R, = 14nm, comparable with the last case, but the QCSE is larger. At 1.064 pm the figure of merit Sa/a - = 4, almost as good as observed in GaAs

118

ELSAGARMIRE

QWs. In an impedance-matched FP, the on-state reflectivity (using fi = 0.8) would be 46%. The maximum absorption loss a+ = 0.5 pm-’ with fifty 10-nm Q W s means that a 60% reflective front mirror could achieve impedance match. Excellent waveguide modulators at 1.06pm using strained InGaAs/GaAs QWs have been reported (Hasenberg et al., 1994; Humbach et a/., 1993), but such studies are beyond the scope of this chapter.

d. 1.3 pm The next challenge is to move the exciton resonance to wavelengths of interest in optical communications. Considerable effort has been undertaken to demonstrate modulation at 1.3 pm, the wavelength at which fibers have zero dispersion, and at 1.55pm, the wavelength at which fibers have minimum loss. The interest of this chapter is in normal-incidence modulators, although most work has been done on waveguide modulators. As early as 1987, QW waveguide modulation at 1.3pm was proposed (Temkin et af., 1987) using 20 lattice-matched InGaAsP quaternary wells of width 10-nm grown on InP. Preliminary results showed a somewhat smeared exciton with a long wavelength parameter I , = 53 nm, presumably due to inhomogeneous broadening connected with crystal growth nonuniformities. Nonetheless, the demonstration of QCSE encouraged further research. Zucker and colleagues explored lattice-matched InGaAsP/InP QWs as waveguide modulators, limiting the number of Q W to reduce the inhomogeneous broadening (Zucker et al., 1989). Photocurrent measurements of a sample with five wells each 7nm wide indicate a long-wavelength parameter ICI= 28 nm and a figure of merit (with 15 V applied) f = 6a/ c( - = 4.3 (at a wavelength of 1.28 pm), with a maximum response of about 40% of the peak excitonic response. Their paper focused primarily on refractive changes at longer wavelengths, for use in waveguide modulators. The use of so few wells limits the application of these lattice-matched quaternaries to waveguides. Strained InAsP ternary wells can be grown on InP to provide modulators at 1.3pm with less compositional inhomogeneity than quaternaries (Hou et al., 1993). Electroabsorption measurements in 10 InAsP wells 10-nm thick indicate a resonance parameter 2, = 20nm. The figure of merit grows rapidly at long wavelength, to a value of 6a/a- = 46 at 55 meV longer than the zero-field exciton resonance. The maximum loss this far from resonance is only a+ = 0.15 p m - I , however, so that it would be very difficult to build a high-contrast normal-incidence modulator with only 10 wells, since the

2 OPTICAL NONLINEARITIES IN SEMICONDU(JTORS

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front mirror reflectivity would have to be impossibly high. Operating closer to the resonance, however, allows 6a/a- = 5 with a 22meV detuning (at a 1.35pm wavelength); here a + = 0.42pm-'. With a front mirror reflectivity R , = exp(-2a+l) = 0.92, an IMFP could be expected to achieve a reflectivity varying from 0 to 53%. More QWs would lessen the requirements on the front mirror reflectivity and produce a normal-incidence modulator less sensitive to tolerances, but strain buildup renders this approach unlikely to succeed. The 10 strained ternary wells offer excellent possibilities for waveguide modulators, however. The growth of more QWs requires removing strain buildup, which can be accomplished on InP substrates by creating an equivalent quaternary QW (Leavitt et al., 1995) using a strained-layer short-period superlattice of two ternaries: InGaAs and InAIAs. Forty wells were grown with 2, = 15nm, clearly superior to the 10-well quaternary results. An optimal figure of merit of 4.8 was reported, with a maximum QW loss a + = 0.23pm-'. An 83% front reflectivity mirror would provide an IMFP modulator whose reflectivity is controllable from 0 to 52% as the internal field varies from 0 to 150 kV/cm. Using the improvements that coupled quantum wells give over the QCSE, researchers at AT&T have found excellent performance at 1.3pm using a slightly asymmetric CQW structure grown on InP (Hou and Chang, 1995). They designed the structure to give chirp-free operation (no electrorefraction) using 25 coupled QWs, with one well of lattice-matched ternary (InGaAs) and barriers of lattice-matched InAIAs. The other well was either a lattice-matched quaternary (InGaA1As), a compressively strained quaternary (InGaAIAs) or a tensilely strained ternary (InGaAs). Both wells were 4-nm wide, but the second had a 30 meV higher band gap. Absorption data indicate an optimal long-wavelength figure of merit of Ga/a- = 4.5 at wavelength 1.38pm, with a maximum QW loss of a + = 0.33pm-' at 79 kV/cm applied field. Using 25 coupled wells with 8-nm well thickness in each CQW pair, the round-trip absorption loss at high field will be 12%, so a front mirror of 88% reflectivity will impedance match an FP, with a predicted high-state reflectivity of 50%. Because of the CQW structure, there is an operating wavelength on the short-wavelength side of the excitonic peak that provides chirp-free operation. Here the absorption decreases with voltage rather than increases. Changes as large as da = 0.55 pm- were measured, but the residual loss (in this case at high voltage) was a- = 0.25 pm - ', so the figure of merit reduced to 2.2. This means that the maximum reflectivity available to an IMFP would be only 27%. Nonetheless, this operating wavelength has the potential for providing a practical modulator because a + = 0.8 pm-' and the required front mirror reflectivity is only 73%.

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ELSAGARMIRE

An alternative explored to achieve normal-incidence electroabsorption is Wannier-Stark localization (WSL). Results were reported in reflection at I .3 pm using lattice-matched InGaAsP wells with InP barriers (Tadanaga et af., 1997). The long-wavelength parameter was as large as A, > 50nm, indicating that residual absorption will limit the figure of merit to below where it would be practical for normal-incidence F P devices. This technique may have advantage for waveguides in materials systems in which the exciton is not well-defined, however. The availability of a variety of material systems, both strained and unstrained, grown on lnP, has already allowed demonstration of practical waveguide modulators at 1.3 pm (Wakita et a/., 1995). Practical normal incidence modulators are sure to follow, when crystal growth challenges are overcome.

Electroabsorption modulators at 1.55-pm wavelengths are challenging because the absorption coefficients in the InGaAs/InP system are only about 40% of those in the GaAs/AlGaAs system. Thus many QWs will be required for practical normal-incidence modulators. The highest quality interfaces and lowest possible doping levels are required. Should ternaries or quaternaries be used? The trade-offs include achieving a sharp-enough excitonic resonance while maintaining a reasonable oscillator strength at high field strengths. The earliest work (Wakita et al., 1987 and references therein) used lnGaAs QW 10-nm thick with InAlAs barriers, lattice-matched to InP substrates, designed to be waveguide modulators operating at 1.55 pm. This work demonstrated speeds approaching 100 picoseconds and 301 contrast ratio, even though the data showed a rather small QCSE shift of 10 meV for 150 kV/cm field. Photocurrent data indicated a rather wide excitonic resonance (long-wavelength parameter I, = 50 nm) in these MBE-grown samples. Larger QCSE shifts and narrower resonances were reported using the same lattice-matched InGaAs wells, but with InP barriers, although this shifts the wavelength of operation out to beyond 1 . 6 p (Bar-Joseph et al., 1987). A Stark shift of 30meV at 150kV/cm with a long-wavelength parameter of R, = 35 nm was reported from 100 Q W of thickness 10 nm grown by MOCVD. Figures of merit as high as f = Ga/a- = 6 were measured for 100kV/cm, with ( a + ) = 0.14pm-',at a wavelength 40-nm longer than the zero-field excitonic resonance (1.61 pm). If the reported 2 pm of material were inserted into an IMFP, the maximum reflectivity would be 57%. using R , = 56%. Thus this material should provide a reasonable

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normal-incidence spatial light modulator at 1.65 pm wavelength. A waveguide modulator 375pm long composed of these QW had 47:l contrast ratio with 3 dB input coupling loss and a minimum of 0.15 dB propagation loss (Koren et al., 1987). At about the same time, workers in England similarly grew latticematched ternary wells (Shorthose et al., 1987). Their data (for 50 wells) show a long-wavelength parameter 1,= 40nm, but it is difficult to infer electroabsorption from their results. Gas-source MBE grew a similar composition (Temkin et al., 1987), but 20 wells had a broader excitonic feature (A, = 55 nm). A waveguide of length 250 pm provided a modulation depth of 35% at 1.64 pm wavelength. Growing quaternary InGaAsP wells pushed the wavelength shorter to 1.55 pm, but resulted in an even broader exciton (A, = 70nm in 20-well samples), due to inhomogeneous broadening from well-width fluctuations. Normal-incidence measurements of lattice-matched InGaAs/InP utilized 150 ternary QWs of thickness 10nm, fabricated by MOCVD (Guy et al., 1988). Transmission in the long-wavelength region (1.64 pm) changed from 52 to 30% with the application of 40 V (130 kV/cm). Taking into account transmission losses from interface reflections, this corresponds to a fieldinduced increase in QW absorption from ( a _ ) = 0.04pm-' to ( a + ) = 0.24 pm-', for a figure of merit 6a/a- = 6, with (6a> = 0.20 pm- '. Near the excitonic resonance (1.59 pm) the transmission increased from 12 to 32% (absorption decreased from ( a + ) = 0.49pm-' to ( a _ ) = 0.17pm-'), corresponding to a much larger absorption change, (6a) = 0.32pm-', but a much smaller figure of merit, f = 6a/a- = 2. The IMFP, which depends on the figure of merit f, clearly prefers long-wavelength operation, in contrast to the simple transmission mode, where short-wavelength operation represented higher contrast ratio because of larger 6a. Using ( a + ) = 0.24 pmand assuming that the front mirror reflectivity is the air/semiconductor Fresnel reflectivity of R, = 30%, 125 QWs would be enough to provide an impedance-matched IMFP. A figure of merit f = 6 implies that this IMFP would have a maximum reflectivity of 56%, confirming that ternary wells with InP are extremely promising for normal-incidence modulators, although their wavelength of operation is fixed by the designed well-width. Normal-incidence measurements near 1.55 pm wavelength were made on InGaAs QWs using lattice-matched InAlAs barriers and InP substrates (Chin et al., 1994). The study of 25 QWs of width 6.1 nm was motivated by their use of waveguides, but also indicates their potential for normalincidence modulators. Because the QCSE shift is smaller, a higher voltage (250 kV/cm) is required to achieve absorption changes comparable to ternary wells with InP barriers (6a = 0.25 pm-'). This occurs at a wavelength of 1.52pm, 17 meV from the zero-field exciton peak, where the

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figure of merit is f = 6a/a- = 7.5. At higher voltages (350 kV/cm) and farther from the exciton resonance (1.533 pm wavelength), 6a was comparable, but with a larger figure of merit, 6a/a- = 20. The narrower wells, however, mean that there is only 0.15% absorption change per Q W (compared to 0.4% for those with InP barriers reported by Guy and colleagues), so a large number of wells are needed before normal-incidence devices can be practical. Lattice-matched quaternary wells with InP barriers, grown by MOCVD for 1.55 pm waveguide operation (Zucker et al., 1989), had photocurrent spectra that indicated broadened excitonic features (A, = 50 nm). Although figures of merit as high as 6a/a5 could be inferred, the serious loss of oscillator strength with applied field in these long-wavelength QWs means that the absorption change per Q W is too small to make an effective normal-incidence electroabsorption modulator unless a large number of wells can be grown, a challenge to crystal growers. Excellent lattice-matched quaternary waveguide modulators have been demonstrated, however (see, for example, Devaux et al., 1993). Finally, Wannier Stark superlattices represent an alternative for waveguide modulators at 1.55 pm (see, for example, Bigan et al., 1992), but are not viable for normal-incidence modulators because of small absorption changes (typically 6a 0.02 pm- '). The ideal material systems are those in which both the well-width and the material compositions can be separately controlled, with barriers high enough that the QW maintain their oscillator strength even when the fields become large. In this important 1.3- 1.5 pm wavelength region, normal incidence modulators need many QW (100 or more) to ensure IMFP without requiring a high reflectivity front mirror. To grow this many quaternary wells lattice-matched to InP challenges crystal growers. Nonet heless, considerable progress is being made toward practical normalincidence spatial light modulators operating at 1.55 pm. The materials that have been optimized for waveguide modulators may not be the most suitable for such normal-incidence modulators, however. The optimization of SLMs for 1.55 pm operation remains to be completed.

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-

7.

ELECTRICALLY CONTROLLED STATE FILLING

In the introduction (as well as elsewhere in this book), state filling was introduced as a valuable mechanism for optical nonlinearities. When carriers remain trapped in a well, they can remove the absorption that otherwise might be there. By electrically controlling the density of carriers in QWs, large changes in absorption and refractive index can be created (Wegener et a/., 1989). This mechanism has been very practical in waveguide

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modulators. However, it appears that it would be difficult to create a sufficiently large filling factor to be practical for normal-incidence modulators, so this mechanism will not be discussed further here.

IV. Experimental Configurations Optical experiments and applications can be carried out in a variety of formats. This section reviews commonly used configurations and derives figures of merit that can be used in each geometry. This section also will compare these models with typical experimental results.

1. TRANSMISSION Transmission experiments are the simplest to perform (see Fig. l), with the transmitted intensity given by I, = I, exp( -aL)

(22)

where the absorption a changes due to the optical nonlinearity. Experiments are usually characterized by two quantities: insertion loss (throughput at high transmission) and contrast ratio (ratio of high transmission to low transmission). Assume that the loss in the QW, initially at high value a+, decreases by 6a to a _ . The low- and high-transmission states are T(10w) = Z,/lo= exp(-a+L) T(high) = l,,/lo= exp(-a-L) where a is the absorption per unit length measured within the QW, and L is the total length of QW material. If there are M wells and L, is the Q W thickness, then L = ML,. As an alternative, sometimes the average absorption in the optical medium is used (which is written here as (a)), and then L is the total length, including the barriers. The contrast ratio is given by CR

= T(high)/T(low) = exp(6aL)

(23)

The insertion loss is given by A

= 1 - T(high) = 1 - exp(-a-L) z a - L

(24)

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ELSAGARMIRE

A long path length L means a high contrast ratio but also a large insertion loss. Choosing the most practical length for any given application, then, requires trading off the contrast ratio and insertion loss. Thin-film transmission geometries are usually grown by epitaxy, with thicknesses around 1 pm, so the nonlinearities must be very large in order for measurable effects to occur in a I-pm thickness. Such effects require direct-band semiconductors monitored very close to the band edge, where the absorption coefficients can be on the order of 1 pm-'. As was seen in Section 111, there are usually two wavelength regimes of large absorption change: (a) on the long-wavelength side of the excitonic resonance, close to the excitonic resonance at maximum static field, where the zero-field absorption is small and the absorption increases with field, (b) close to the zero-field excitonic resonance, where the absorption is high but can decrease with increasing field. Either regime may be used in transmisson experiments. (The signs and relative magnitudes given here are for QCSE and FranzKeldysh effect; they will change if coupled QWs or Wannier Stark localization is used.) u.

Operation at Longer Wavelengths

On the long-wavelength side of the zero-field exciton resonance, the loss decreases rapidly with increasing wavelength. In the QCSE, the exciton resonance moves to a longer wavelength with increasing static field but also decreases in peak absorption. As a result, there is an optimal wavelength that is a function of static field and there is an optimal static field at which the absolute absorption change is a maximum (shown in Section 111). In this section I will use the results of such an optimization to model expected performance for GaAs/Al,,,Ga,,,As Q W modulators, using the values shown in Table I. Optimization of 6a (Jelley et al., 1989) occurs for a QW thickness L, = 3.5 nm at an internal field of 240 kV/cm and a wavelength such that a - =0.43 pm- ' for the low-loss state and a + = 2.68 pm- ' for the high-loss state. Then a 150 QW sample, with a total QW thickness of 0.525 pm, would yield 6aL = 1.2 and a contrast ratio of 3.3:l when used in transmission, with a high transmission of 80% and a low transmission of 24%. The contrast ratio can be improved by using twice the path length (by placing a mirror behind the nonlinear film and operating in reflection), yielding a contrast ratio CR = 11:l with a high transmission of 64% and a low transmission of 6%. It is worth pointing out that the transmission contrast ratio is optimized by optimizing GaL,, not 6a. Analysis of the data (Jelley et al., 1989) shows that optimized values of the product 6aL, are essentially constant over well

125

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS TABLE I

OFTIMAL ELECTROABSORFTION VALUESDETERMINEDFROM EXPERIMENT FOR GaAs WELLS AND 10-nm Alo.33Gh.6,AsBARRIERS.'

L,(nm) ahh(pm-') a- ( p m - ' ) 6a(pm-') Max. 6a/aMax. 6aL Max. SaL Resonance

3.5 6.5 8.5 3.5

4 3 2.5 4

0.43 0.41 0.38 2.2

2.2 1.7 1.3 -1.7

6aL,

6u/u-

0.008 0.011 0.011 0.006

5.2 4.1 3.4 -0.8

E,,,(kV/cm) 240 120 80 286

The first line represents optimal figure of merit J: The second and third lines represent optimal long-wavelength absorption change and the last line gives values at the excitonic resonance wavelength. 'From Jelley et al., 1989.

widths from 6.5 to 8.5 nm, and do not vary more than a factor of two over the entire range of well widths from 3.5 to 12.5 nm. Some typical values are shown in Table I. What is important in design is that the maximum field be chosen optimal for a given well width, which specifies the wavelength of operation. Result. Simple transmissive films may be useful in low-contrast applications, but require optimized QCSE because GaL must be maximized and L is limited by crystal growth considerations. Furthermore, there is always some residual transmission in the off state, requiring thresholding detection for practical applications. Improved performance is achieved by using a double pass through the Q W epilayer.

b. Operation on Zero-Field Exciton Resonance Typical data (Jelley et al., 1989) show that the absorption measured at the exciton resonance in a 3.5-nm QW will decrease with increasing field from a+ = 4.0pm-' to a- = 2.2pm-', yielding Ga = -1.8pm-'. To achieve a contrast ratio of 3:l using this absorption change would require 160 QWs, with a high transmission of 30% and a low transmission of 10%. Doubling the path length would improve the contrast ratio to 9 1 but results in a high transmission of only 9% and a low transmission of 1%. Result. Operating a transmission modulator on the excitonic resonance is inherently lossy, even for modest contrast ratios; long-wavelength operation yields a better contrast ratio for simple QCSE. However, the shortwavelength region is important for SEED devices, which require an increase in absorption with photoinduced screening of the applied field.

126 c.

ELSA GARMIRE

Waveguide Operation

Alternatively, waveguides with lengths on the order of 100pm or more can be used, typically on the long-wavelength side of the excitonic resonance, where the loss is relatively low. Operation on resonance in waveguides requires using a small degree of optical confinement so that the average loss is acceptable in the longer lengths. To keep a moderate insertion loss, waveguide lengths must be chosen so that L 5 l / ( a - ) , where ( a - ) is the loss averaged over the waveguide: ( a _ ) = Fa-, where is the filling factor of the QW in the waveguide. Inserting the length condition into the contrast ratio,

CR

= exp(ba/a-)

(25)

The figure of merit for the largest possible contrast ratio in a waveguide is therefore the fractional change in absorption, defined with respect to the low-loss state. The data (Jelley et al., 1989) show that for QCSE in Q W s of thickness L , < lOnm, optimal ba/a- occurs under the same conditions as optimal 6a. Peak values reported for 6a/a- are 5.2, so CR = 181 is achievable in waveguides. (This is not achievable at normal incidence because the condition L z l / a - would require 2.3 pm of QW material, or 660 QWs, many more than have been grown successfully.) With enough QWs in a waveguide to provide a QW filling factor r = 2.3%, a waveguide length L = 100jim provides (a-)L= 1, for a net transmission of 37%. Under high loss, when a + = 2.7 pm- (a+)L = 6.2, for a net transmission of 0.2%. yielding a CR of 185.

’,

2. ABSORPTION-ONLY INTERFEROMETER The contrast ratio in an electroabsorptive transmissive device is ultimately limited by the inevitable residual transmission at high loss. In principle, improvement can be found by using interference to cancel this residual signal. This requires using a reference beam I, to interfere with the voltage-varying signal beam I,. Then the output, measured under conditions of destructive interference, would be

where the two intensities are chosen such that they will become equal when the field increases the loss, so that the output will be zero, the “off-state”.

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

127

The “on-state” will occur under low loss, for which I, = I, exp( -6aL). This change in absorption produces an output in the on-state given by

Because the off-state now has zero transmission, the contrast ratio, by definition, is infinite. The important quantity, then, is the transmission in the on-state, which depends quadratically on GaL/2. Assume the best case, where 6 a L is as large as possible in normal incidence (150 QWs optimized at longer wavelength). Then GaL = 0.14, which means that the maximum throughput would be 25%.

Result. Interference can be used to achieve an arbitrarily large contrast ratio in an electroabsorption modulator, but only at the expense of throughput in the on state.

3.

INTERFEROMETER

BASEDO N

PHASE S H I n

In general, whenever there is an absorption change, there is also a refractive index change, which can be measured and/or used in an interferometer. To change from destructive to constructive interference, the phase must change by a, which requires (Gn)kL 2 n, or L 2 4(2(6n)). Typically, in the wavelength regime below resonance where the losses are lowest, <6n) 5 0.01. For R = 820 nm, achieving a phase shift would require QW material of total length L w 40 pm, clearly impossible at normal incidence. If used on resonance (and ignoring loss), 6n can be as much as 0.1. But even in this case, 4pm of QW material would be required, which is unrealistic. The alternative is to use either a Fabry-Perot (discussed below), where a smaller phase shift is required to move off resonance, or a waveguide, where the length can be longer. With waveguides, the insertion loss must be kept moderate. Specifying an insertion loss as A = 1 - T(high) sz (a-)L determines the maximum length L. Since L = A / ( a - ) , an interferometer requires

Typically A x 1/2. This interferometric criterion is often difficult to achieve near the band edge because of residual loss. Thus waveguide interferometric modulators typically operate well out on the long-wavelength side of the

128

ELSAGARMIRE

excitonic resonance. An early example of an interferometric waveguide modulator is given by (Zucker et al., 1988).

4.

FABRY-PEROT GEOMETRIES

The interaction of the nonlinear medium and the light intensity can be enhanced by using a Fabry-Perot etalon (FP) that consists of mirrors on the front and back of the semiconductor epitaxial layer (see Fig. 21). Interference of light within the FP allows for higher contrast ratios with smaller nonlinearities compared with transmission devices. The contrast ratio can be made arbitrarily large by proper design, even when the fractional change in transmission is small. In a reflective FP, light reflected off the front mirror interferes with the light that is reflected back and forth within the slab. This interference is constructive or destructive depending on the round-trip phase change within the FP. Zero net reflection results from destructive interference when the reflectances of the two mirrors are properly chosen and matched to internal losses. Then the F P is called impedance matched (Siegman, 1986). When the loss inside the FP changes, the amplitudes of the two reflected components are no longer equal, there is less destructive interference, and the FP reflectivity increases from zero. a.

Introduction

Since a properly designed impedance-matched Fabry-Perot (IMFP) can always show complete destructive interference in its “off-state,” with zero

rL

I

_3

i

R1,

FabryPerot FIG. 21. Geometry of Fabry-Perot etalon and Bragg mirror showing incident I,,, reflected RI,, and transmitted TI, beams.

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

129

reflectivity, the design challenge is to maximize the “on-state” reflectivity, defined when the reflectivity is a maximum. When only the absorption can change, there are two possible F P configurations:

i. Impedance matching (R = 0) in the low-loss state, with the FP reflectivity switching on when the absorption loss becomes high; the FP reflectivity increases because introduction of large loss removes the destructive contribution from the back mirror, and the FP reflectivity occurs predominantly from the front mirror only. ii. Impedance matching (R = 0) in the high-loss state, with the FP reflectivity switching on as the absorption loss is reduced. The FP reflectivity increases because, as the loss decreases, reflected light from the back mirror will tend to overwhelm that from the front mirror.

I will show that both configurations offer the same “on-state” reflectivity when properly designed. However, case 2 requires a smaller front mirror reflectivity and is thus easier to fabricate. In fact, the air-semiconductor interface alone may provide enough reflectivity to create a suitable impedance-matched cavity. Impedance match will occur for either zero applied field or maximum applied field depending on configuration and on the operating wavelength. It was shown earlier that there are two wavelengths at which the QCSE is particularly large: (a) on the long-wavelength side of the resonance, where a field increases the absorption, and (b) near the exciton resonance, where a field reduces the absorption. I will show below that the best performance occurs where the ratio of absorption change to residual absorption is largest, which is usually case (a). b. Modeling the Reflective Fabry-Perot

The reflectivity of a Fabry-Perot, in general, is given in terms of the half-round-trip phase, 4, the front and back mirror reflectivities R, and R,, respectively, and the loss a by (Garmire, 1989)

R=

+ F sin24 1 + F sin2$

R,,

where F=

4Re (1 - RJ2

ELSAGARMIRE

130

and

The effective reflectivity is

On resonance when 4 = mn, the FP reflectivity R = Rmx;this is the case when destructive interference minimizes R . Impedance match occurs when destructive interference ( 4 = mn) causes R = 0. This requires an asymmetric Fabry-Perot with Re = R Varying the loss inside the resonator changes Re, thereby destroying the impedance match and increasing the reflectivity. The absorption loss can either increase or decrease depending on configuration, which offers two cases for consideration. in case 1, the loss increases away from impedance match. From Eq. (32) it can be seen that, as the loss increases, Re decreases, tending to zero in the limit of infinite loss. In this limit, Eq. (31) predicts the on-state reflectivity Ron-,R,, since the back mirror is hidden by the loss. Case 2 assumes that the loss decreases away from impedance match. From Eq. (32 it can be Seen that R , will increase from R, at impedance match UP to in the limit of no loss and R , = 1. In this limit, Eq. (31) predicts Ron 3 1. Expressing the front mirror reflectivity in terms of the loss at impedance match. RJi = R , exp( - 2a,L)

(33)

A subscript is added to R , and o! to indicate that it is the absorption occurring at impedancematch (zero reflectivity); this is the“off state” of the Fabry-Perot. When a changes, destructive interference is no longer complete, and the FP reflectance increases from zero. The Fabry-Perot is no longer impedance matched this is the “on state.” Assume that the new absorption z differs from a, (either larger or smaller). Then Re has a new value, Re = R, exp( - r,L ) exp( - aL). Also, R,/R,i = exp( + o!,L)exp( - EL). The Fabry-Perot reflectivity in the “on state” will be

Ron

= R,i

{ 1 - exp[ - ( a ( 1 - R,exp[-(a

- a,)L]}’

+ a,)L])’

(34)

where the “on state” assumes absorption change only, under the phase condition of destructive interference, 4 = mn. Let us compare case 1, where

2 OPTICAL NONLINEARITIES I N SEMICONDUCTORS

131

a >> a,, and case 2, where a << a, when the front mirror reflectivity R, is chosen for perfect impedance match.

1. In case 1, the impedance match occurs under low-loss conditions, so a, = a-, while the “on state” operates under high-loss conditions, with a = a+. Writing 6a = a - a, and inserting Rli = Rbexp(-2a-L) into Eq. (341, Ron= R, exp( - 2a - L)

[I - exp( -6aL)]’ (1 - Rbexp[-(6a + 2a-)L])’

case 1

(35)

or

’

[1 - exp( -SaL)] Ron = 1 ‘ [l - R, exp( -6aL)I’

case 1

As the absorption change gets very large, Ron--* R,. That is, the back mirror is completely blocked, and reflection occurs only from the front mirror. 2. In case 2, impedance match occurs under high-loss conditions, so a, = a+, whereas the “on state” operates under low-loss conditions, with a = a _ . Writing 6a = a, - a and inserting Rli = Rbexp(-2a+L) into Eq. (34),

Ron = R,e~p(-2a+L)

[I - exp( +6aL)]’ (1 - Rbexp[-(6a Za-)L]}’

+

case 2

(37)

Using the fact that a + = 6a + a- in the exponential in the first factor in Eq. (37) and then moving the factor exp( -26aL) into the square, it becomes clear that this is the same expression as Eq. (33, case 1, except for a sign change within the square. It is useful also to write Eq. (37) as

’

[1 - exp( + GaL)] Ron = R, [1 - R, exp( GaL)]’

+

case 2

To understand what this means, the maximum that 6aL can be is to reduce the loss to zero: exp(-6aL) = exp(-a+L) = Inserting this into Eq. (38) gives the maximum that Ron can be, which is the lossless case

&/A.

132

ELSAGARMIRE

The highest contrast ratio will occur when the back mirror reflectivity can be chosen to be 100% (by using, for example, an integrated quarterwavelength stack as a Bragg mirror, as discussed below). In this case, Ron -+1, a result that is expected because the F P would be lossless under these conditions and the back mirror is 100% reflective. The only difference between case 1 and case 2 is that the impedance-match requirement on the front mirror reflectivity leads to a much lower reflectivity in case 2. By varying the number of QWs in the FP (varying the value of L), one has control over the cavity loss and therefore over the required front mirror reflectivity. Thus, in principle, the impedance-match requirements for either case can always be fulfilled. In practice, however, there must be sufficient loss that the front mirror is not unrealistically high in reflectivity. Let me give a specific numerical example of a case where the front mirror consists of the air-GaAs interface and the back is a 100% reflective Bragg mirror (Parry et al., 1991). In order to achieve impedance match with the air-GaAs reflectivity R, = 0.3 (case 2), the product of the high-loss absorption coefficient and the Q W length should be such that a + L = -ln(R,)/2 = 0.6. Electroabsorption data at 6 V give a maximum a, = 1.2 pm- which would require 50 wells of width lOnm to achieve the impedance-match criterion. The data show the residual low-loss coefficient a- = 0.2pm-’, so that the 50 wells of Parry et al. (1991) should give a - L = 0.1, resulting in GaL = 0.5. Inserting these numbers into Eq. (38) gives Ron = 0.5, a FabryPerot reflectivity of 50%, in reasonable agreement with the reported ”on-state” reflectivity of 45% (Parry et al., 1991), although the details of voltage and the number of Q W s differ in specific, possibly because of the characteristics of the underlying Bragg reflector. The authors reported a high contrast ratio modulator (1oO:l) that was impedance matched at 9 V (high loss) and had a low-loss reflectivity of 45% (the “on state”). If these QWs were used in case 1, with impedance match at low loss, the reflectivity of the front mirror would have to be 82% instead of the 30% that was used in case 2. Thus the easiest case to build is when impedance match takes place under the high-loss condition. Destructive interference is balanced between the front mirror and the attenuated back mirror. As the loss is reduced, the back mirror begins to strongly dominate the F P reflection, and the overall reflectivity goes up. In the limit where the loss is reduced to zero, clearly the F P must tend to 100% reflectivity when the back mirror is assumed 100% reflective, because there would be no other source for loss. When long-wavelength electroabsorption is used, which has low loss at zero applied voltage, case 1 is traditionally called “normally or’, since 0 V represents zero reflectivity, whereas case 2 is traditionally called “normally on,” since 0 V represents high reflectivity.

’,

2

OPTICAL

NONLINEARITES IN SEhWONDUCTORS

133

Figure 22 shows two examples of an asymmetric Fabry-Perot spectrum under impedance-matched conditions, when the loss increases from impedance match and when the loss decreases from impedance match. Both cases assume that most of the loss can be removed so that aL goes from 0.06 to 0.6. In case 2, the impedance match occurs under the high-loss condition, and R, = 0.3. As the loss decreases and the FP reflectivity increases, the finesse of the cavity stays low because of the low-reflectivity mirror. In case 1, impedance match occurs under the low-loss condition, and a front mirror reflectivity of 89% is required. The finesse is very much higher under this impedance-matched condition, leading to considerably more stringent conditions on temperature and alignment. Note also that the “on-state” reflectivities are the same for the two cases. Result. Case 2 is preferred, requiring a lower front mirror reflectivity to achieve the same overall reflectance change within the FP, from zero to the reflectivity in the on state and operating with a low-finesse cavity. In the ideal case, where loss can be removed completely, the reflectivity changes from zero to R, = 1. For a high contrast ratio, it is important to achieve the

Case ii)

Wavelength

Case i)

Wavelength FIG.22. Spectrum of asymmetric Fabry-Perot modulator showing zero reflection at impedance match with increasing reflectivity for either greater or less absorption loss. Case 2 a+L = 0.6 at impedance match and a - L = 0.06 at high-reflectivity, with R , = 0.3. Case 1 assumes a - L = 0.06 at impedance match and a + L= 0.6 at high reflectivity, with R, = 0.89. The wavelength scale is the same. In both cases the reflectivity increases from 0 at impedance match to 66%.

134

ELSAGARMIRE

design requirement R, = exp( -2a+L) and R, + 1. On the long-wavelength side of the exciton resonance this represents the “normally-on” electroabsorption modulator. I‘.

Finding Figures of Merit

The designer has control of the mirror reflectivity needed to achieve impedance match, since he or she can always (in principle) alter the thickness to ensure that R, = Rbexp(-2a+L). Thus a zero-reflectivity “off state” can always be chosen. How does the on-state reflectivity depend on the nonlinear optical material? To understand this, write GaL = Pa+L, where fi is the fraction of absorption that can be removed as the field changes. Then Eq. (38) (case 2) becomes

However, since exp(a+L) =

Ron

A/&,Eq. (39) can be written as

= R,

[I - (R f/Rb)-8’2]2 ( I - R,(R,/R,)-@’~)~

where fi = 6a/ci+ can take any value from 0 to 1 (at which point no loss remains). Numerical evaluation of Eq. (40) in the limit 1 - R, << a+L indicates that this expression is essentially independent of R,, as shown in the curves of Fig.23. The surprising result that the reflectivity in the “on state” of an I MFP controlled only by changing absorption is essentially independent of R , can be derived analytically, giving the result that, to second order in the Taylor expansion,

where fl = ba/ci+, f = 6a/6a- and R , = 1. Note that /? = f/(l + f). In order to show that Ron is essentially independent of R,, make the substitutions R , = exp( - 2a+L)and R, = 1:

2 OPTICAL NONLINEARITIES IN SEMICONLWCTORS 6

1

135

I

e .. b

Y

d

Y WY

k 0

Fractional absorption change: t?

= &/a+

FIG. 23. Reflectivity of an impedance-matched asymmetric Fabry-Perot in the “on state,” as a function of the fractional absorption change for varying reflectivities of the front mirror R, (which equals the round-trip absorption loss under impedance-matched conditions). Note that the curves are essentially independent of R,. Triangles represent an analytic fit: Ron = [/?/(2 /?)I2. where /? = 6a/a+.

-

The exponentials can be expanded in a Taylor series, keeping their values to second order:

Note that the a + L factors cancel out, and the product in the numerator can be expanded to ( 1 -a+L)(l /?a+L/2) = 1 - a + L + Ba+L/2, which cancels the denominator. Thus, to first order, = -/?/(2 - p), deriving Eq. (41). The triangles in Fig. 23 represent this approximate solution. Table I1 gives typical values for the reflectivity as a function of varying fractions of absorption decrease. It can be seen that unless B = 6a/a+ > 0.5 (and f > l), the on-state reflectivity will be so small as to be unreasonable. In fact, to achieve 50% reflectivity in the on-state requires the field to remove at least 83% of the loss. It is interesting to note that the amount of absorption change is not the driving consideration here; it is fractional change in absorption. This optimization differs from that for single- or double-pass transmission through a thin film. This also means that those papers which report only absorption change cannot be applied directly to F P device analysis because

+

136

ELSAGARMIRE TABLE I1

FALIRY-&ROT ON-STATE REFLECTIVITYFOR VARYINGFRACTIONS OF ABSORJTION DECREASE R.3” 11 = 6 a k ,f = 6a/z

+

0.1

0.2

0.4 0.77

0.6

0.62

0.3 0.71

0.5

0.5

0.83

1

1.63

2.4

3.3

4.9

0.87 6.7

0.7 0.91 10.1

0.8 0.94 15.7

0.9 0.97 32

1 1 99

knowledge of the baseline absorption is also required. The form of these equations also shows that using many QWs (increasing L) does nor show up as an improved figure of merit; what is important is the ratio Ga/ci+. lncreasing L is, however, important in reducing the required R,. A peak value off in GaAs/A1,.,,Ga0,,,AsQWs is given as 5.2 (Jelley et a/., 1989). This means that the largest /3 can be is 0.84. Referring to Table 11, the largest expected FP reflectivity in the “on-state,’’ produced by pure absorption modulation in these QWs, would be on the order of 50% if the rear mirror had 100% reflectivity. This is close to experimental results (Parry er al., 1991).

d

Eflect of Refractive Index Change

The analysis so far has been for electroabsorption only, ignoring any possible refractive index change in the FP. To analyze the effect of refractive index change, note that the reflectivity is changed by the parameter F sin2d, where Q, is half the round-trip phase change. For small phase shifts away from an integral number of z due to a refractive index change, define 4’ = 4 - mz = Gn2zL/1. Then the reflectivity changes from its value at mx to (assuming F$2 << 1)

R=

R,, 1

+ F$”

+ F412

x R,,

+ F412(1 - R,,,,)

where F and R,, were given by Eqs. (30) and (31). This indicates that refractive contributions that change the phase always increase the F P reflectivity from the value it has at complete destructive interference (4‘ = 0). The increase in reflectivity in the ”on-state’’ due to a refractive index change from the “off-state” will be

6R

%

F&’(l - R,J

where we can write F as

F=

4exp(-aL- a,L) [l - exp( -aL - a,L)]’

(43)

2 OPTICAL NONLINEARITIES IN SEMICONDUC~ORS

137

Note that F is independent of whether case 1 or 2 is being used. Thus the refractive index contribution will be the same in either configuration. For the numerical case given above, (a -L = 0.06, a, L = 0.6), F = 8.8 and R,, = 0.66. (Note that the absorption in either the “off’ or “on state” will be large enough that the oft-used expansion of the exponential will not be valid.) In the assumed case, where L = ML, = 0.5 pm and I = 0.82 pm, then 2nL/1=3.8. Thus 6R = 44 6n2. If it were possible to operate right on the excitonic resonance, then 6n can be as large as 0.1 and 6R can be significant. However, in the band tail, where 6n I0.02, the change in reflectivity due to refractive contributions is much smaller, 6R I0.02. e. Experimental Examples of Semiconductor Etalons The first examples of a surface-normal Fabry-Perot etalon for nonlinear optics were explored for optical logic elements (Gibbs et al., 1979). It was shown, both experimentally (Poole and Garmire, 1985) and theoretically (Garmire, 1989; Wherrett, 1984), that optical bistability with a higher contrast ratio is achieved by operating an IMFP in reflection. Considerable research continued on optical logic elements using band-filling semiconductor nonlinearities in a Fabry-Perot etalon. The ability to grow integrated mirrors by single-crystal epitaxy directly on the nonlinear film made particularly elegant devices (Sahlen et al., 1987a; Jewel1 et al., 1987; Gourley and Drummond, 1987). The Fabry-Perot in reflection has proved to be a valuable electrically driven spatial light modulator. The first MQW electroabsorption modulator integrated with mirrors was demonstrated in reflection (Boyd et al., 1987; Guy et al., 1988, 1989b; Simes et al., 1988). Rapid developments between 1988 and 1991 found increasingly high contrast ratios (Whitehead et al., 1989; Yan et al., 1989; Goossen et al., 1989; Law et al., 1990; Pezeshki et al., 1990). Contrast ratios of 1OO:l with “on-state” reflectivity W 4 0 % were barriers. among the better results in GaAs QW with Al,~,,Ga,,,,As New materials were investigated, such as InGaAs QW with Al,,,,Ga,~,,As barriers as IMFP modulators (Pezeshki et al., 1991). Reflection changes as high as 77% were reported in an FP structure with 5 GaAs/AlAs layer-pairs as a front mirror (R, 80%) and 18.5 layer-pairs as a back mirror (R, 99%). The FP contained 30 wells with 20% indium in the QW, for a peak contrast ratio of 371 at a wavelength of 1.026pm. An effective IMFP modulator has even been made with an epitaxial bulk GaAs film 1.9-pm thick, operating on the Franz-Keldysh effect, lifted off its substrate and placed between reflective silver and a reflective I T 0 contact. A contrast ratio of 75:l was reported, with 32% reflectivity in the “on state” with 20 V applied (Tayebati, 1993).

-

-

138

ELSAGARMIRE

Simulation has been developed to optimize asymmetric Fabry-Perot modulators (Lin et al., 1994b; Neilson, 1997), although no dramatically better performance has resulted, perhaps in part because of nonuniformities occurring during crystal growth (Goossen et a/., 1995b) and the difficulties in achieving the optimum impedance-match condition (Timofeev, 1995). Investigations of ways to make such modulators more practical continue (Trezza et al., 1993; Prucnal et a/., 1993; Zumkley et af., 1993). Some directions for research are the extension of the electrically switched FabryPerot to other practical wavelengths, such as 1.06, 1.3 and 1.55 pm. Detailed discussion of the development of the asymmetric Fabry-Perot reflection modulator is beyond the scope of this chapter. However, experimental results are consistent with the modeling done here, where reflectivities in GaAs MQW modulators can be changed from 0 to -40%. A recent example shows close to theoretical performance for asymmetric Fabry-Perot M Q W modulators at low incident optical power and discusses their degradation at high optical power (Mottahedeh et al., 1994). Nonlinear Fabry-Perots with MQW hetero-nipi material placed between two reflectors enable optical control of reflectivity from 0 to 40% (Larsson and Maserjian, 1991a), as will be discussed in Section V.

f.. Fubry-Perot Operated in Transmission The Fabry-Perot can be used in transmission also. The transmission of a lossy Fabry-Perot can be written as (Garmire, 1989) T=

H 1 + Fsin2&

(45)

where

H

= (1

- R,Xl - Rb)exp(-aL)/(l - Re)*

The transmission and contrast ratio can be compared to the simple transmissive film most simply when it is assumed that R, = Rb E R. The transmission in the low-loss state is given by

Tmax =

(1 - R)* exp( -a-L) [l - Rexp(-a-L)]’

The contrast ratio between the high-transmission and low-transmission

2 OPTICAL NONLINEARITIES IN SEMICONDU(JTORS

139

state is given by CR =

exp(GaL)[l - R exp(-a+L)I2 [l - R exp(-a-L)I2

For large loss in the lossy state, exp(-a+L) << 1. Then, CR =

exp(6aL) [l - Rexp(-a-L)]’

Note that this is the transmission contrast ratio times an enhancement As factor over the transmissive thin film given by [l - Rexp(-a-L)]-’. with the simple transmissive film, there is a trade-off between the contrast ratio and the transmission in the low-loss state. The best performance occurs when a-L << 1 - R. Then CR x

Tmax =

[l

exp(6aL) (1 - R)’

exp( -a- L) - R)]’

+ a-L/(l

= 0.06. When R = 40%, T,, = 78% and CR = 4.8. If, however, R = 8O%, CR increases to 43 but T,,, reduces to 55%. Nonetheless, the use of an FP increases CR by many times over single pass transmission in the latter case. The transmissive FP will have use for applications in which the reflective geometry is awkward. Transmission modulators have been shown to be useful for interconnects and have been demonstrated by several authors (Calhoun and Jokerst, 1993; Fritz et al., 1993; Lin et al., 1995;Trezza et al., 1996). Contrast ratios as large as 7.4:l have been reported, with 30% transmission in the “on state.”

As an example, consider GaL = 0.54 with a-L

5. BRAGGMIRRORS Bragg mirror is the term generally applied to dielectric multilayer mirrors grown integrally on an active device (modulator, laser, or detector). The Bragg mirror consists of pairs of quarter-wavelength layers with alternating high and low refractive index to provide high reflectivity. One of the first GaAs/AlGaAs Bragg mirrors, grown by MOCVD (Van der Ziel and Ilegems, 1975; Thorton et al., 1984), reported that 40 layers with

140

ELSAGARMIRE

A1,,,Ga0~,As alternating with 40 layers of GaAs gave a peak reflectivity of -98%. With larger concentrations of aluminum, smaller numbers of quarter-layer pairs are needed. Pure AlAs ( n = 3.0) and GaAs ( n = 3.6) dielectric steps can give 98% reflectivity with only 15 layer pairs. Techniques have been developed to oxidize the AlAs layers, making the required number of layers even less, since the aluminum oxide refractive index approaches 1.5. These mirrors are valuable for VCSELs (vertical cavity surface-emitting lasers), with reflectivities as high as 99.95 and 99.97% predicted from mirrors grown with seven interior pairs and five outside pairs (plus an air interface), respectively (MacDougal et al., 1996). a.

On-Resonance Reflectivity

A simple expression for the on-resonance reflectivity (assuming that each layer has a thickness of a quarter of an optical wavelength) is found in optics textbooks (Born and Wolf, 1980). When there are N pairs of quarterwavelength layers that alternate high index and low index ( n , and n,, respectively), and n f and ni are the refractive index of the final and initial media, respectively, the reflectivity is given by the square of a simple reflection coefficient:

h.

Of-Resonance Refectivity

Off resonance, when layers are not the correct thickness and/or when loss is included, matrix multiplication can be used to provide the correct reflectivity (pp. 55-60 in Born and Wolf). I show here a simpler formulation of the matrix technique that we have developed that is particularly efficient in numerical computation (Ramadas et al., 1989) because it enables use of predominantly real transfer matrices. Consider a structure consisting of N layers in the xy plane (which may be lossless or absorbing), sandwiched between two semi-infinite media, as shown in Fig. 24. For a wave propagating at an angle C#I with respect to the z direction, with a plane of incidence xz, the electric field (for TE polarization) or the magnetic field (for T M polarization) in the y direction can be expressed as a linear combination of two linearly independent solutions. These are expressed as forward and backward waves in Born and Wolf, with the running waves resulting in imaginary terms in the transfer matrix of each

141

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

TRANSMllTED

no

ni

n2

n3

...

nN

nf

<-<--

A0 60

A1 B1

A2 62

63

... ...

AN EN

Bf -->REFLECTED

Af <--- INCIDENT

FIG. 24. Geometry for real matrix evaluation of Bragg mirrors. Diagram shows position d , refractive index n,, and field coefficients A,, B,, for each layer. Light is incident from the right and is either reflected or transmitted.

layer. I suggest there is advantage in using a standing-wave formalism so that the transfer matrices are real. In the ith layer, of refractive index ni, the field polarized in the y direction can be written in real terms (if there are no losses) as

Ei = Aicos[k,(z - di)]

+ Bi<,sin[ki(z - di)]

(47)

where A, and Bi are arbitrary constants, k, is the longitudinal (2) propagation constant k, = nik, cos 4i, where 4, is the angle of propagation in the ith layer, di is the position coordinate of the beginning of the ith layer, and t i 3 l/ki for TE polarization and 5, = n:/k, for TM polarization. Note that k? = (k2,nf - k:), where k, is the (continuous) transverse propagation constant and k, = 2n/R,. The boundary conditions require that A, and Bi must be continuous across the ith boundary. Applying these boundary conditions leads to the transfer analysis for multilayer films:

where the transfer matrix for the ith layer is

si =

cos Ai -(l/ti)sin A,

ti sin Ai cos Ai

and Ai = ki(di+ - di). Let the total overall matrix have the result

1

(49)

142

ELSAGARMIRE

In the lossless case the matrix elements will all be real. The coefficients A and B at the beginning and end of the film are related by: A,

=

M,,A,

+ MI2BO

and

Bf = M,,Ao

+ M,,B,

(51)

To determine the solution, impose the condition that only outward-going waves can exist at the exit face of the multilayer structure. It is most convenient to define layer 1 as the lust layer and the Nth as the j r s r layer that light experiences. Then the requirement of only outward-going waves after the last layer, in the geometry of the figure, means that there is only a backward wave. The forward wave would have the general form of ( A , - iB,<,)(exp @), where @ describes the phase. For this wave to be identically zero requires A, - iB,<, = 0 or A, = iB,<,. The incident wave, backward going in the region z 4 d,, will be

and the reflected wave is

The reflection coefficient divides the amplitude of the reflected wave by the incident wave:

or A-B r=A+B

(55)

where A = i M , , < , + M , , andB=(iM,, +M,,(,)
from which the transmissivity can be calculated through T = ((,/4,)ltl2. These equations give the same reflection and transmission coefficients as Born and Wolf but enable a faster calculation using real transfer matrices

2

OPTICAL NONLlNEARlTlES IN SEMICONDUCTORS

143

for lossless media; only the final equations are complex. Loss or gain can be included by using complex refractive indices and propagation constants.

c. Result of Coupled-mode Theory Coupled-mode theory gives comparable results to the matrix evaluation, (Kim and Garmire, 1992) particularly in the regime of small refractive index change between high- and low-index layers ( nH and n 3 , such as typically used in semiconductor Bragg gratings. Coupled-mode theory permits an analytic expression for the frequency dependence of the reflection coefficient. When there is absorption in the high-index layers only, the reflection coefficient for light of wavelength Iz is r=

i(K - iF) scoth(sL.) + i(S - iA)

(57)

where the coupling coefficient due to index modulation is

K = sin(an)

nH2 - nL2 nA1

+

with a = nL/(nH nL) and nA = 2n,n,/(nH coefficientdue to absorption modulation is

+ n,),

u sin(an)

F=-

27c

whereas the coupling

(59)

Also, the detuning parameter 6 = k, - n/A is expressed in terms of the optical momentum change 2k, by subtracting off the effect of the grating periodicity, 27c/A. The normalized absorption A = a(a/2), and s is given by s’ = ( K

- if‘)’

+ ( A + id)’

(60)

This analysis assumes that the substrate and superstrate have a refractive index that is the average of that in the two layers. The detailed shape of the reflectivity as a function of wavelength depends on whether air or another semiconductor region is placed adjacent to the quarter-wavelength stack and can be added to the coupled-mode analysis in a straightforward manner (Kim and Garmire, 1991). A plot of a typical reflectance spectrum in the presence of differing amounts of loss is shown in Fig. 25.

144

Wavelength FIG. 25. Typical Bragg reflector spectrum for varying amounts of loss, with the reflectivity decreasing from the lossless case and developing more asymmetry with increasing amounts of loss.

d. Fabry- Perot Resonators Using Bragg Mirrors Bragg reflectors have been very useful in monolithic nonlinear optical devices. In the usual configuration, transparent Bragg reflectors are inserted between a nonlinear film and an opaque substrate, with the film used in reflection without removal of the substrate. Very often a n additional reflector is provided on the top surface of the nonlinear film, creating a Fabry-Perot interferometer. This top reflector may be the Fresnel reflection of the top air-semiconductor interface, it may be a metal or dielectric mirror, or it may be another semiconductor Bragg reflector. The nonlinear resonator, then, consists of a nonlinear film sandwiched between a Bragg reflector and an additional mirror. Some of the experimental results of such structures, particularly for electrically driven modulators, were described above in the section on the Fabry-Perot. The first integrated Fabry-Perot modulator that included epitaxially grown Bragg mirrors used QCSE in a p-i-n structure (Boyd et al., 1987). The electroabsorption reflection modulator consisted of 65 QWs 9.7 nm thick with 6.6-nm barriers of A1,,,,Ga0,,,As in the i layer grown on a Bragg reflector consisting of 12 pairs of AlAs and AI,,,Ga,,,As, each a quarter optical wavelength thick. The contrast ratio was -8:1, with peak reflectivity of 25% at 853 nm. Operating voltage was 0 to 18 V.

2

OPTICAL

NONLINEARITIES IN SEMICONDUCTORS

145

The first nonlinear Fabry-Perot devices that used MQWs and an integrated Bragg reflector mirror (Sahlen et al., 1987b) demonstrated optical bistability in reflection with 30 mW of input light, a result comparable with what had been seen previously with etalons constructed by removing the substrate and separately fabricating dielectric or metal mirrors on the epitaxial “flake.” The structure of Sahlen et al. had a Bragg reflector quarter-wavelength layers consisting of 20 pairs of A1As/A1,,Ga0.,As between the MQWs and the substrate. The top mirror was fabricated after growth by a deposited dielectric multilayer stack. Section V will describe in more detail some nonlinear Fabry-Perot resonators that use integrated Bragg mirrors coupled with carrier transport nonlinearities.

e.

Nonlinear Bragg Reflector

Another geometry utilizes a nonlinear Bragg reflector, which occurs when one (or both) layer in the quarter-wavelength stack has a nonlinear absorption and/or refractive index. The first nonlinear Bragg reflector required very high powers (30 MW/cm2) and operated on the plasma nonlinearity (Gourley et al., 1987). The first nonlinear Bragg reflectors using the band-filling nonlinearity were demonstrated in GaAs/AlGaAs grown by MOCVD (Kim et al., 1989). In this case, only the high-index quarterwavelength layer was nonlinear (the low-index layer, with a larger band gap, was transparent). The band-filling nonlinearity changed primarily the absorption and therefore the reflectivity of the Bragg reflector from 47 to 85% at power levels of 57 kW/cm2. Yet another embodiment places a half-wavelength film between two quarter-wavelength stacks. This is the proper geometry for a “notch” interference filter, a narrow-band transmission device. Such a nonlinear Bragg structure was investigated in ZnSSe (Karpushko and Sinitsyn, 1978), although in multicrystalline evaporated film and not in a single-crystal structure. The electrical modulation of light transmitted through multilayer interference films consisting of GaAs/AIAs (Yoffe et al., 1987) used the FranzKeldysh effect to change the refractive indices of the GaAs. A strong applied electric field (200 kV/cm) was needed to shift the transmission edge enough to modulate the intensity of light tuned to the edge of the high-reflectance band. Modulation ratios for transmitted light of up to 2 5 1 were reported, but required voltages on the order of 100 V. The periodicity of the nipi structure lends itself to consideration of using it as a Bragg reflector (Kost et al., 1988b) that can be modulated optically

146

ELSAGARMIRE

or electrically. The first experimental nipi Bragg reflector (Goossen et al., 1990b) was electrically controlled by means of selective contacts to the doped regions. The structure contained a half period of GaAs and a half period of AIAs, with delta doping at the beginning of each half period. The reflection changed from 33.6 to 38.2% as the voltage changed from - 1 to I V. This small reflectivity change was attributed to the linear electrooptic effect because the wavelength of operation (1.064 pm) was far from the band edge. The Bragg hetero-nipi as a nonlinear optical device was reported (Poole er af., 1994) to have a reflectance of 7% that changed to 18% on application of 2 mW/cm2. This carrier transport device will be discussed in more detail in Section V.

6. FOUR-WAVE MIXING

Another way to achieve high-contrast nonlinear optical performance is with four-wave mixing. This geometry uses two pump beams near normal incidence but with a small angle between them. The interference between these two pump beams forms a standing wave inside the semiconductor. The nonlinearity creates a grating in absorption and refractive index that can be used to diffract a probe beam into a new direction. This is a background-free measurement because there is normally no light in the direction of the diffracted beam, and this technique has proven practical for a number of applications, such as image processing. In many cases, a separate probe beam is not required, because the interference of the two pump waves can cause self-diffraction. However, a separate probe beam can be time delayed, offering information on time dependence. Several other chapters in this volume have considerable discussion of four-wave mixing, so it will not be considered further here, although four-wave mixing experiments have been useful in developing an understanding of carrier transport nonlincarities.

V.

Characteristics of Experimental Devices that Utilize Self-Modulation

This section reviews some experimental characteristics of self-modulation nonlinearities used in optical devices and then looks at some of the reported performance of the type I and type I1 hetero-nipi’s when used in nonlinear optical devices. As has been true throughout this chapter, the devices are designed to be used at normal incidence, optimized for high sensitivity and

2 OPTICAL NONLINEARITIES I N SEMICONDUCTORS

147

seeking high contrast, for applications such as optically addressed spatial light modulators (Maserjian, et al., 1989; Larsson and Maserjian, 1992).

1. SPEEDOF nipi STRUCTURES From the first introduction of the nipi structure, it was clear that the recombination time of electrons and holes was vastly slowed down by their spatial separation (Dohler, 1983). As photocarriers move to screen the internal field, the potential barrier separating the carriers decreases, and recombination becomes more probable. As a result, the photocarrier lifetime decreases with photoexcitation. Early measurements of the frequency response of the optical nonlinearity (Ando et al., 1989b; Maserjian et al., 1989; Garmire et al., 1989) indicated that this would be an important characteristic that explains the strong saturation of the available absorption change. A model of the recombination time that has proven useful depends exponentially on the potential barrier between electrons and holes (Dohler et al., 1986):

where 70 is the recombination time for electrons and holes located at the same point in physical space, k is the Boltzmann constant, T is the absolute temperature, and V is the potential barrier experienced by photocarriers and is given by

where Ed is the dark field (in the absence of light), Q is the photoinduced two-dimensional camer density that has separated to screen the dark field (in units of carriers/cm2),and E is the static dielectric constant. The distance d is the average distance between the electrons and holes before they recombine. After picosecond pulse excitation, d is the distance between the n and p regions, where carriers in the dark collect. In the presence of CW illumination, however, this distance is much shorter than the distance between the n and p regions because of the presence of a steady-state distribution of constantly replenishing photocarriers throughout the i region (Yang et al., 1997). Experimental data on type I1 hetero-n-i-p-i structures indicated that d is typically the distance from the doped region to the nearest QW. Since Q depends not only on the applied optical intensity but also on the lifetime, the preceding equations must be combined to determine the

148

ELSAGARMIRE

dependence of lifetime on optical intensity, as will be shown in the model described in the next subsection. The measured recombination time has indeed been shown to depend exponentially on the two-dimensional carrier density (Yang et al., 1997). Light of intensity I near the band edge will create a three-dimensional carrier density within the QW given by N = a l ~ / h vwhere , a is the absorption coefficient in the QW,hv is the photon energy, and r is the carrier recombination time. If there are M quantum wells in a given i region, absorption will create a total of MaL,Ir/hv carriers per square centimeter. If these carriers separate enough to screen a fraction / of the wells, on average, then the separated charge density is

Q =j M a L , l r / h v

(63)

Experiments reporting the dependence of measured lifetime on CW intensity showed, as expected, that the response time decreases very rapidly with increasing intensity (Simpson et a/., 1986; Garmire et al., 1989; Ando et al., 1989a; Law et al., 1989; Kost et al., 1989a). Equation (63) suggests that measurements of I and T can be combined to indicate Q as a function of intensity (when changes in a are small). That is, IT as a function of intensity will give an indication of how rapidly the carrier density (and therefore the screening field) increases with intensity. The results of pure nipi’s (Simpson et al., 1986) indicated that IT a 1 1’3. Experimental data on type I heteronipi’s (Kost et al., 1989a) could be fit by IT cc I”’. More recent studies of delta-doped type J MQW hetero-nipi’s (Jonsson et al., 1994) also can be fit to IT c€ I I Q . By comparison, most results on type I1 nipi’s show that the product of 17 is essentially constant, even over four orders of magnitude in intensity (Garmire et al., 1989). This means that increasing the intensity causes an absorption change at a much slower rate than the intensity increase. That is, most of the intensity increase goes into compensating for decreased carrier lifetimes. Typical recombination times are on the order of milliseconds for pump intensities on the order of milliwatts. In order to have practical devices, it is necessary to speed them up, which can be done by pixelation and the provision of conductive paths for carrier recombination (Ando et al., 1989b). A single type 11 hetero-nipi device of dimensions 2 x 1.5 mm was cleaved, and a thin gold layer was provided by vacuum evaporation on the cleaved edges of the sample and annealed at 500°C to form an ohmic contact with both the n and p layers. A comparable sample without the ohmic contact also was fabricated for comparison. Ando and colleagues found that the response was speeded up, from an original value (at 1 pW/cm’) of 20 ms to

2 OPTICAL NONLINEARITIES I N SEMICONDUCTORS

149

0.5 ms at 1pW of incident power, while the magnitude of the absorption change decreased from an original value of 6a = 3000cm-' to 6a = 160cm- The devices with shorted contacts had a recombination time that was essentially independent of pump level (0.3 ms), whereas the unshorted devices had a lifetime that decreased from 60 to 10ms as the pump power increased from 0.3 to 3 pW. Meanwhile, the absorption change was essentially constant with intensity for the unshorted devices (6a = 3000 cm-'). By contrast, but with shorted contacts the absorption change increased by 20 times as the intensity changed by 20 times. Thus, with shorted contacts, a !z 1/Z, as expected for devices with a constant lifetime. Of course, if devices were pixellated on the scale of micrometers, speeds could be pushed to nanoseconds.

'.

Result. While nipi devices have inherently slow recovery times, this can always be speeded up by pixelating and shorting the n and p layers, to produce a constant recombination time.

2.

MODELING TYPEnipi STRUCTURES

The performance of nipi structures as nonlinear optical materials is strongly affected by the dependence of carrier recombination time on the internal field within the device. Modeling shows that it can take as much as 12 orders of magnitude of intensity, in principle, to screen the entire internal field and reach saturation. However, in type 11 hetero-nipi's, even before this can happen, decreasing the internal field far enough can cause the energy of the electrons in the QW to become lower than the energy of the electrons in the doped regions. At this point, carrier transport ceases and state filling in the first available QW begins. a.

Simple Model for Intensity Dependence of Absorption

This subsection presents a simple model for the intensity dependence of the self-modulation nonlinearity within a nipi. The result will explain why absorption changes larger than 6a = 7000cm-' are difficult to achieve in type I1 hetero-nipi's. The independent variable in the model will be the two-dimensional photocarrier density Q that has been separated by carrier transport. As shown recently (Yang et al., 1997b), under CW excitation, this carrier density is a heuristic quantity that averages the local variation in carrier density throughout the band-bending region and is useful because it allows

150

ELSAGARMIRE

the rate equation to be used. Writing an equation for the electrons that have separated along z, dQ/dt = G - Q/r

(64)

where the generation rate G = alfML,/hv. This assumes that a fraction t of the total number of Q W in an i layer, M, have contributed carriers that have separated to produce the charge sheet density Q. The Dohler model for carrier recombination time will be used:

where d, is the distance carriers must travel to recombine. In the dark, this distance will be the thickness of the i layer. However, under CW illumination we showed that this distance will more typically be the distance from the doped region to the first Q W (Yang et al., 1997b). Experimental results showed lifetimes that fit the preceding equation in a typical type 11 hetero-nipi if d, was taken to be on the order of 15 to 30 nm. Self-modulation comes from the screening of the internal field by the transport of photocarriers. For Q carriers per unit area separated, the internal field changes from its value in the dark Ed to

Self-modulation occurs because the change in internal field will change the absorption by QCSE (see Section Ill). As a model for the field dependence of absorption through QCSE, I write the absorption coefficient as a function of the two-dimensional photocarrier separation Q through the internal field

where g o , a,, and 0 1 ~are experimentally measured values. Treating Q as the independent variable, we can invert Eq. (63) to write the incident intensity in terms of the two-dimensional photocarrier separation that it causes, as well as the recombination time and absorption:

=

Qhv T(Q)a(Q)JML,

For any given nipi, it is necessary to know the internal field. This requires

2

OPTICAL

NONLINEARITIES IN SEMICONDUCTORS

151

knowing the amount of band bending due to the difference in the Fermi levels between the n- and p-doped regions. As an example, consider doped regions consisting of Al,Ga, -,As whose band gap is given by (Chuang, 1995) WB,(X) =

1.424 + 1 . 2 4 7 ~

(69)

The p region usually will be nondegenerate, while highly doped n regions will be degenerate, and Fermi statistics must be used to find the Fermi level, Nonetheless, the assumption of nondegeneracy does not introduce significant error for WB, (3% here). The approximate thickness of the depletion layer within the n region can be calculated from Poisson’s equation through:

where Li is the length of the intrinsic region separating the p and n regions. There is a similar equation for the thickness of the depletion layer in the p region, using N,, the doping density of acceptors. The internal field in the nipi structure comes from the charge that leaves the depleted layer and migrates across the i layer. The amount of this charge determines the internal field, and it is important to determine whether or not the doped regions are fully depleted. The n-doped region will not be fully depleted as long as d, < LD/2, where L, is the width of the doped region. In this case, the total two-dimensional charge that initially migrates across the i layer is

When this inequality is violated, the proper equation for the initial charge that migrates is

Thus, in the dark, the internal field in the nipi is

where y = 1 if the doped layers are not fully depleted and y = (LD/2)/d, if the layers are fully depleted. If the p layer is fully depleted, y = (LA/2)/d,,. We now have all the equations necessary to calculate the absorption as a function of incident intensity. First calculate the dark field (Eq. 73). Then

152

ELSA GARMIRE

specify the separated photocharge Q, which allows the internal field to be calculated (Eq. 66). Then both the lifetime and absorption will be known (Eqs. 65 and 67). This allows Eq. (68) to determine the intensity that caused this separated photocarrier density Q (and this absorption a and this lifetime 5 ) . From these numerical values it is now possible to plot the absorption change as a function of intensity. The result is shown in Fig.26. The numerical values used are those appropriate to the experimental results of Yang and Gannire (Fig. 10): x = 0.3, N, = N, = 1018/cc,with the doped regions not fully depleted and Li = 124 nm. It also was assumed, for simplicity, that a. = 0, a , = 0, and a,EZ = 1 x 104cm-', which is fit to experimental measurements. It can be seen from the figure that five orders of magnitude increase in intensity are required to remove half the excess absorption. To fully saturate

-5

I

log intensity

5

log intensity

5

4

2

u L

c '2 c

.-0

B a

4 CD

3

2

-5

FIG. 26. Calculation of absorption change as a function of intensity in a t y p I1 hetero-nipi, including the carrier density dependence of the internal field and the recombination time. Note that it takes 10 orders of magnitude of intensity, in principle, for the absorption change to saturate, i.e.. for the bands to completely flatten. Over many orders of magnitude in the midrange. the absorption is essentially linear with log of the intensity. Therefore the log-log plot may be more useful for estimating saturation values. The marked half-absorption point determines the saturation intensity. On the log-log plot, this point is 0.3dB smaller than the muration value.

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

153

the absorption requires an additional five orders of magnitude in intensity! Thus the experimental results that demonstrate between 15 and 25% change in transmission, but rarely more, are explained by the rapid decrease in carrier lifetimes with intensity. This effect is compounded by the quadratic dependence of the absorption on field, which means that the absorption changes more slowly as the field becomes smaller. It also can be seen from the figure that over a large range of intermediate intensities, the absorption change is linearly proportional to the log of the intensity, which is the experimental result most often noted. The difficulty of predicting a saturation intensity from measurements in this linear regime can be observed; a log-log plot is more useful, as shown in Fig. 26b. From the saturation value of the absorption change, the saturation intensity can be found, since log(aJ2) = log(a$ - 0.3. The saturation intensity is shown on the plots. An additional effect occurs before the bands are fully flattened. As the internal field decreases, the quantized energy of the electrons in the QW becomes lower, until it falls below the energy of the electrons in the doped region. At this point, carrier transport ceases and state filling begins. In the case of Yang’s sample a 40-nm spacer layer separated the first 8-nm QW from the doped region. The QW is 374meV deep, with the first level lying 55 meV above the GaAs band edge. The Fermi level in the doped region is calculated to be 8nm above the band edge, and the potential rise to the first QW due to the internal field (before any screening) is (45/127)Eg = 578 meV. Thus, by the time the screening has reduced the internal field by half, the QW nearest to the n region will begin to fill. Higher intensities, reducing the field further, will allow additional QWs to fill. The filling of the bands offers its own contribution to a change in absorption with increasing intensity. The existence of band-filling nonlinearities at higher intensities in hetero-nipi’s was observed as an additional increase in transmission as the incident intensity became very strong (Kost et al., 1991), and this was confirmed by picosecond studies (McCallum et al., 1991). Thus it is reasonable to assume that the self-modulation nonlinearity in a type I1 hetero-nipi will typically operate with the internal field decreasing no more than a factor of 2 when exposed to light.

b. Simulations of the Optical Nonlinearity in nipi’s The absorption saturation shown in the simple model above was confirmed by a complete model for the saturation of the nonlinear absorption in nipi structures containing a single QW in the i region (Xiaohong and Jin, 1995). The calculation showed a saturation level for the built-in field of approximately 80 kV/cm, determined by the balance of photogenerated

154

ELSAGARMIRE

carriers moving into the doped region and their removal by recombination processes that include radiative recombination as well as Auger recombination for the lightly and heavily doped layers. Xiaohong and Jin pointed out that the saturated value of the built-in field will be reached when the bottom of the Q W is lower than that of the conduction band edge of the doped layer. On further illumination, photocarriers will collect in the QW instead of in the doped layers. A simulation of carrier recombination in a type I delta-doped M Q W structure (Jonsson er a/., 1994) found intrinsic recombination rates calculated for ideal structures far below those found experimentally. Jonsson and colleagues found that the dependence of the measured lifetime on the optically induced excess carrier density behaved as if the effective amplitude of the space charge potential was three times lower than expected. Their analysis assumed that recombination could take place only by intrinsic recombination between carriers in the two delta-doped layers (radiative tunneling recombination and spatially direct recombination of thermally excited carriers). On the other hand, if it is assumed that some photogenerated carriers are always in the i regions under CW excitation, the effective distance that comes into the Dohler formula for carrier recombination time is reduced by the photocarrier density that must exist in the i layer during steady state. While this density may be very small, it can have a dramatic effect on the measured response time because recombination time depends exponentially on this value (Yang et al., 1997b). 3. PICOSECOND EXCITATION OF nipi's With pulsed excitation, the photocarrier recombination time does not matter, as long as the pulse is short compared with whatever recombination time the nipi experiences. Screening half the field should require twodimensional photocarrier generation equal to half the initial Qi calculated above. The fluence of pulses required to screen half the field is

F, = IT

= (Q/2)(hv)/(aLi)

(74)

For the nipi decribed above, where the charge transfer from the doping regions is Qi = 1 x 10'2/cm2and aL,= 0.124, this gives a screening fluence of F, = 0.9 jd/cm2. This fluence should remove approximately three-quarters of the absorption because of the quadratic nature of QCSE below the band edge. Studies of charge transport using picosecond pulses with fluences of between 1 and 10 pJ/cm2 (McCallum, 1991) illucidate some of the phenom-

2 OPTICAL NONLINEARITIES IN SEMICONDUCTORS

155

ena underway on the picosecond time scale in type I1 hetero-nipi’s. The samples investigated were all-binary InAs/GaAs using short-period strainedlayer superlattices within the QW. Using 2-ps pulse excitation with a fluence of 1 pJ/cm2, state filling is manifest by the time the pulse is over, showing up as a pure bleaching of the exciton resonance. This gives way within 6 ps to a blue shift of the exciton as the carriers have escaped the wells and begun their drift to the doped regions, screening the built-in field. The signal rises to 90% of its maximum value in 12 ps. On the other hand, experiments with 10 times this pump fluence observed predominantly state filling, since the carrier density in the QW swamps any screening field carrier density. Other experiments with even faster pump pulses (0.6 ps) and a single Q W in each intrinsic region (Moritz et al., 1995) demonstrated the cooling of hot electrons followed by thermionic transfer of the thermalized carriers out of the Q W to the doped regions. This transfer time increased from 50 to 500 ps with increasing excitation power, due presumably to a slowing of the drift velocity as the internal field decreased.

-

4. LATERAL ENHANCEDDIFFUSION Because the lifetimes are so long in nipi structures, carriers live long enough to diffuse significant distances laterally. This was observed in the earliest experiments (Ianelli et al., 1989; Ando et al., 1989). The effect of lateral diffusion in n and p layers on the response time of p-i-n diodes was observed in the picosecond regime (Livescu et al., 1989) and modeled successfully in terms of the capacitance, based on simple electrical theory, and termed “enhanced diffusion.” This model was found to agree with experiments describing picosecond response of p-i-n MQW structures both electrically (Schneider et al., 1992) and optically (Yang et al., 1994, 1997). Similar concepts were applied to nipi structures, with calculations compared with measurement (Thirstrup et al., 1990). When photocarriers are generated by a small-diameter laser pump beam, the excess majority carriers will drift radially out from the beam region and recombine. They produce a lateral voltage drop because of the finite sheet resistance of the neutral regions. Measurements of the decrease in absorption change as a function of the separation between the pump and probe verified this picture and provided an estimate of lateral hole mobility of 425 c m 2 / V - s . Experiments that separated the pump and probe by a distance of more than 10 spot sizes depressed the absorption change by only half its maximum value. Further studies of enhanced diffusion in nipi’s analyzed it in terms of a giant ambipolar diffusion constant (Gulden et al., 1991; Gulden et al., 1992), pointing out that the driving force for the diffusion of carriers in the lateral

1%

ELSAGARMIRE

direction is no longer solely the gradient of the carrier density. There is a very strong additional contribution due to the repulsive Coulomb potential, which is proportional to the lateral gradient of the carrier density. It is this fact that allows a giant ambipolar diffusion constant to be defined. The following ambipolar diffusion coefficient has been derived (Gulden, 1991; Streb 1997):

where Qn, Q p , pn, and pp are the two-dimensional densities and mobilities of electrons and holes in the n and p layers, respectively, and cP is the differential capacitance density. A p-i-n structure was fabricated, and the lateral diffusion was measured (Streb et af., 1997). Excellent agreement was found between theory and experiment. Measurements were made in the picosecond domain of enhanced ambipolar in-plane diffusion (McCallum et al., 1993) and the preceding equation was used to compare with experiments of the diffusion as a function of injected carrier density. The experimental results confirmed this model for enhanced diffusion. Clearly, lateral diffusion is a dominant phenomenon in nipi's and practical devices must be pixelated to remove it.

5.

EXPERIMENTAL PERFORMANCE OF nipi's INSERTED INTO DEVICES

In GaAs/AlGaAs devices, the QW resonance is at a higher energy than the band gap of the GaAs substrate, necessitating one of two approaches to device design. Either the substrate must be removed, or a Bragg reflector must be placed between the nonlinear film and the substrate so that the device can be used in reflection. With long-wavelength devices, such as InGaAsP/InP or strained InGaAs/GaAs, the substrates are transparent, and this is not a consideration. Whether or not the substrate is transparent, a much greater contrast ratio as a modulator can be achieved by inserting the nipi's in an impedance-matched Fabry-Perot and operating in reflection. This can be done in an integrated fashion by growing a Bragg mirror on top of the substrate before the nipi structure is grown. With proper design, impedance match can be achieved with the air-semiconductor interface as the only front mirror. The alternative structure that has been investigated is a nipi nonlinear Bragg mirror.

2 OP~ICAL NONLINEARITIES IN SEMICONDUCTORS

a.

157

Type 1 Rejective Fabry-Perot Utilizing a Bragg Mirror

A high contrast optically addressed modulator was fabricated in the type I periodically delta-doped InGaAs/GaAs MQW structure shown in Fig. 27 (Larsson and Masejian, 1991). The nipi was monolithically integrated into an asymmetric Fabry-Perot resonator consisting of an underlying Bragg reflector (144 pairs of AlAs/GaAs) and an air-GaAs interface, designed in such a way that the QWs were always located at the peak of the standing wave of the etalon. In order to create the largest possible filling factor, there were two QWs per n-doped region. Periodic doping planes of 1.7 x 10” n type (Si) were centered between the QW pair, and 5.1 x 10” p type (Be) doping planes were placed halfway between the n planes. The InGaAs QWs were each 6.5nm thick separated by a 10-nm GaAs barrier. The distance between the n and p doping planes was 61.5 nm and there were 33 i regions (and a total of 33 QWs). Optical experiments were performed with a low-power InGaAs/GaAs QW laser to generate the write (pump) signal. A contrast ratio of more than 601 with an “on-state” reflectivity of 34% was measured at the highest write signal power used (30 mW). The design was planned for fabrication into an array of etched pixels for optically addressed spatial light modulation, although this was not done. Top reflector: Jr/G&s cavity: &doped

Bottom reflector: InGaAdGaAs MQW AIAdGaAs DBR

a

FIG.27. Epitaxial geometry for a type I hetero-nipi Fabry-Perot structure grown on a Bragg mirror with quantum wells located at the maxima of the optical standing wave. (After Larsson and Masejian, 1991c.)

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The authors inferred that the absorption essentially followed the expected logarithmic dependence on excitation intensity. Some additional saturation observed at higher intensities was attributed to electron rejection from the QWs into the adjacent spacer layers. The maximum absorption change was inferred from modeling the experimental data to be 6a z 9000cm- with a write signal power of 30mW/cmz (taking into account sample size and diffusion).

',

h. Nipi in a Rejlective Fabrv-Perot Utilizing a Bragg Mirror

A pure nipi was inserted into an asymmetric Fabry-Perot (Hibbs-Brenner et al., 1994), and its performance was measured. The GaAs nipi had 75-nm i regions placed between doped layers 9 nm wide. This structure was grown 2.5pm thick on a 15-period AIAs/Al,~,Ga,,,As Bragg mirror with an air-semiconductor interface as the front mirror. In a similar sample, a long-wavelength (average) absorption change of ( 6 a ) = 870 cm- was measured with 10 mW of power. In their Fabry-Perot experiments, HibbsBrenner and colleagues measured a reflectivity change from 20 to 63% with powers up to 100mW, from which they inferred that the absorption changed from ( a + ) = 860cm-' to ( a - ) = 180cm-' for a figure of merit of Sa/cr.. = 3.8. These numbers predict that a n IMFP would have a maximum reflectivity of 44%. This is close to the measured reflectivity change. The reported lifetime was 25 p s for pump powers greater than 4 m W and 6 0 p s at 1 m W .Their sample size was 1 cm', and they observed a 15% reflectance change for 5 mW. If this were pixelated and put into an IMFP, it could become an optically addressed SLM. If the pixels were 10pm on a side, there would be lo6 in a square centimeter, and a 5-mW beam would be able to process a million pixels simultaneously in speeds of 2511s. This has promise for improvement over the liquid-crystal SLMs that are typically in use now.

'

c.

Tvpe 11 Hetero-nipi in a Reflective Fabry-Perot Utilizing a Bragg Mirror

A Franz-Keldysh type I1 hetero-nipi was inserted between an air/GaAs front mirror and a Bragg back mirror to act as an asymmetric Fabry-Perot (Hibbs-Brenner el aI., 1994). This structure contained thirty i layers of pure GaAs (no QW) 200nm thick with n and p layers of AI,,,Ga,,,As of thickness 10 nm. The on-off contrast ratio varied from 5:l at 1 mW pump power to 601 at 9 mW pump power, illuminating a sample of area 1 cm'. The lifetime was 60 p s at 1 m W and 25 p s for pump powers greater than 4 mW. The required power to achieve the 601 contrast ratio was three times lower than in the previously reported type I asymmetric Fabry-Perot.

2 OPTICAL NONLINEARITIES M SEMICONDUCTORS

159

d. Nonlinear Bragg Reflector A proposal to combine the periodicity of Bragg reflectors and doping superlattices was made in 1988 (Kost et al., 1988) and demonstrated (independently) 4 years later (Poole et al., 1992). The original proposal calculated that a 2% change in refractive index by optical modulation would lead to a 2% change in the coupling coefficient and a change in reflectivity from zero to nearly 100%. The device was designed to operate on changes in refractive index and not on changes in absorption. The type I1 nipi Bragg reflector that was reported experimentally utilized nonlinear absorption. There were four coupled quantum wells in each i layer, with a total of 23 i layers. Photoconductivity measurements confirmed optically mediated electroabsorption by a shift of the band edge in the presence of the pump. The authors reported a change in reflectivity from 12 to 16% with the application of up to 3.2 mW/cm2. The fractional change in reflectivity was comparable with other hetero-nipi structures that did not use Fabry-Perot geometries to enhance the contrast ratio. The authors noted, as others had previously, the existence of enhanced diffusion, which showed up as an independence of the result on the size of the pump beam or on the lateral position of the probe with respect to the pump over the 1-cm2 sample. Further studies of a type I1 hetero-nipi reflection modulator were carried out using a single QW in each i region (Poole et al., 1994). Poole and colleagues measured an optically mediated change in reflectivity of 0.11 and a maximum contrast ratio of 4.41 at 1.9mW/cm2 pump intensity. Their numerical model of the device predicted reflectivity modulation of 0.53 and a contrast ratio of 25:l in an optimized device structure. Even this case, however, has an inability to achieve a zero reflectance state using absorptive Bragg structures, which limits their usefulness. e. Prospects for Practical Applications

The results of Larsson and Maserjian (Larssen and Maserjian, 1991c) allow for a scaling to predict SLM performance. If pixels were 10pm on a side, they would have an area of 10-6cm2. Then one square centimeter could be illuminated by 30mW and, according to the results of Larsson, control lo6 pixels with a contrast ratio of 601, enough pixels to be interesting for image processing. Of course, this low control power would require that the pixelation maintains the same response time. The response time of these devices was “in the millisecond range,” a time constant that is roughly comparable with, or perhaps slightly faster than, liquid-crystal

160

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displays. If the response time were speeded up, the intensity needed for illumination also would have to be speeded up. Suppose that a 1-W laser were available; then, by suitable design and pixelation, the response time could be pushed to a few microseconds over an entire 1O0Ox 1000 array. This is most definitely an improvement over available technologies and indicates that research should continue. A comparable analysis can use the results of the GaAs pure nipi SLM (Hibbs-Brenner, et a/., 1994). The optical power was 6 times smaller and the devices were 50 times faster, so this structure is considerably more promising as far as sensitivity is concerned. If it were pixelated without losing sensitivity, and if it could be put into an impedance-matched FP with the same driving power, a 5-mW beam would be able to process a million pixels, 10 pm on a side, simultaneously, in 25 ps. The challenge is to achieve a reasonable contrast ratio in the nipi without increasing the required optical power. If this can be done, then speeding up the pixel response to microseconds should be possible while still keeping the driving power below 200mW. The implication of the experimental studies to date are that the carrier transport nonlinearities are on the verge of practicality. Their decided advantage is that carrier lifetimes can be varied through device design anywhere from milliseconds to nanoseconds. The longer lifetimes mean higher sensitivity (as low as microwatts), although there will be some optimum in the lifetime-sensitivity tradeoff for any particular application. The drawbacks that must be overcome for the carrier transport nonlinearities are the strong intensity dependence of the lifetime and the prominence of enhanced lateral diffusion. Both drawbacks can be overcome by pixelating the samples. Once done, scaling laws predict that impedance-matched Fabry-Perots should provide reasonable competition to liquid-crystal and magnetooptic spatial light modulators.

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SEMICONDUCTORS AND SEMIMETALS. VOL. 58

CHAPTER3

Ultrafast Transient Nonlinear Optical Processes in Semiconductors D. S. Chemla DARTMENT OF PHYSICS, UNIVERSITY OF CALIFORNIA AT BERKELEYAND MATERIALS Scmvca DIVISION. LAWXENCEBERKELEYN A T ~ N ALABORATORY L BERKELEY, CALIFORNIA

1. INTRODUCTION . . . . . . . . . . . 11. NEAR-BAND-GAP EXCITATIONS. . . . 111. TIMESCALES AND DYNAMIC TRENDS.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .

IV. A PURELY COHERENT PROCES INVOLVING ONLY VIRTUAL ELECTRON-HOLE PAIRSTHEE X C I ~ N IOFTICAL C STARKEFFECT . . . . . . . . . . . . V. FUNDAMENTALS OF TWO-PARTICLE CORRELATION EFFECTSINVOLVING REAL . ELECTRON-HOLE PAIRS . . . . . . . . . . . . . . . . . . . . . v1. APPLICATIONS: SPECTROSCOPY AND DYNAMICS OF ELECTRONIC STATES IN . HETEROSTRUCWRES. . . . . . . . . . . . . . . . . . . . . . VII. FUNDAMENTALS OF FOUR-PARTICLE CORRELATION EFFECTSINVOLVING REAL . ELECTRON-HOLE PAIRS . . . . . . . . . . . . . . . . . . . . . VIII. DYNAMICS IN THE QUANTUM KINETICSREGIME . . . . . . . . . . . . . IX. CONCLUSION. . . . . . . . . . . . . . . . . . . . . . . . . LIST OF ABBREVIATIONS AND ACRONYMS. . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Introduction The electronic ground state of a perfect semiconductor is pretty boring. In these materials, electronic energy levels are well described by the efective mass approximation; the electronic ground state is such that the valence bands are full, the conduction bands are empty, and essentially nothing happens. Things become more interesting when this peaceful situation is disturbed; i.e., defects are present, lattice vibrations interact with the electrons, or an external perturbation is applied. This chapter will mostly be concerned with the perturbations caused by an ultrashort laser pulse whose photon energy lies in the vicinity of the fundamental absorption edge. In this *List of Abbreviations and Acronyms can be located preceding the references to this chapter.

175 Copyright 'CI 1999 by Academic Press. All rights of reproduction in any form reserved. ISBN 0-12-7521674 ISSN 0080-8784199 $30.00

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case, the material is raised from its ground state into an excited state, which can be described in terms of electronic excitations whose structure results from a delicate balance between quantum statistics and the Coulomb interaction. The fundamental quasi-particles, electrons (e) and holes (h), obey Fermi statistics. They can form more complex objects with an internal structure, such as excitons (X),biexcitons ( X , ) , polaritons (X-hv),Fanoresonances, dressed holes at the origin of Fermi edge singularities, e-h plasmas, etc. These objects are essentially delocalized and long-lived. Obviously, their nature and properties are very sensitive to the density and energy of the photons that create them. Despite the underlying Fermi statistics of the basic constituents e and h, some composite quasi-particles, X and X , , can, in very specific low-density regimes, exhibit a Bosonic behavior. A further critical dependence on the density stems from the fact that the strong and long-range Coulomb force that is responsible for the formation of these composite excitations can be screened, in particular by e-h plasmas, thus changing their very nature. All these manybody mechanisms are modified in a nontrivial manner by quantum confinement in heterostructures or under high magnetic fields. Just on creation, the excitations are coherent, their quantum mechanical phase being determined by that of the laser light that generates them. Very quickly, however, scattering within the electronic system and with the phonons starts to destroy that phase, and the system evolves toward thermodynamic equilibrium, first among the electronic excitations and then later with the lattice. This fast and complicated kinetics has an impact on fundamental issues related to coherence, dephasing, dissipation, and memory, thus raising some interesting questions about the validity of approximations well established in the quasi-static regime. The combination of extreme density dependence and ultrafast kinetics makes the behavior of electronic excitations in semiconductors and their heterostructures a fascinating topic at the frontier of condensed-matter physics. Their creation or destruction is associated with interband and intraband polarization waves, which, in turn, determine the optical properties of the material. The recent developments in ultrafast laser techniques have provided new tools, perfectly adapted to the study of semiconductor optics, thus opening new opportunities for investigating manybody effects in correlated quasi-particle systems in regimes previously inaccessible. A wealth of valuable and novel information on the physics that governs the electronic excitations of semiconductors has been obtained over the last decade, explaining the spectacular growth of the field of nonlinear spectroscopy of these materials. Coherent nonlinear optical processes were first investigated in atomic and molecular systems using continuous-wave (CW) and long-pulse lasers (Levenson, 1982; Shen, 1984; Mukamel, 1995). In these systems, the energy

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levels are narrow and usually well separated, so the experiments were well described by theoretical models involving only a few levels. In particular, in the case where one optical transition is nearly resonant with the exciting photons and all others are far away, the two-level atom model very successfully accounts for most experimental results in the frequency domain (Allen and Eberly, 1987), and the time domain (Yakima and Taira, 1979). Similar coherent nonlinear spectroscopy techniques in the nanosecond regime were applied to semiconductors for studying excitons (Kramer et al., 1974; 1976) and biexcitons (Maruni and Chemla, 1981; Chemla and Maruani, 1982). Here again, two- or three-level models were sufficient to explain the most salient results, giving more support to these phenomenologic approaches. With the development of short-pulsed lasers, picosecond-time-resolved experiments were performed on semiconductors and heterostructures (Hegarty et al., 1982; Chemla et al., 1984). Coherence and dephasing, however, continued to be discussed in terms of atomic systems concepts using formalisms such as the two-level atom model (Schultheis el al., 1985; 1986a). As the quality of the samples and the laser performances improved, experimental observations were found to be in qualitative contradiction with the “atomic” pictures. This has triggered a major reconsideration of the theory of the dominant mechanisms governing ultrafast processes in semiconductors,which is still being improved. Recently, several excellent review articles and books on light-semiconductor interactions have been published. However, they tend to concentrate on experimental (Shah, 1996) or theoretical (Haug and Koch, 1993; Binder and Koch, 1995; Haug and Jauho, 1996) aspects of the subject. It is my opinion that the spectacular progress made in the field of ultrafast nonlinear spectroscopy of semiconductors is the result of the cross-fertilization between experiment and theory. The parallel developments of advanced ultrashort-pulse laser sources and very sensitive data-acquisition techniques, on the one hand, and formal theory and sophisticated numerical simulation, on the other, has resulted in a new and extremely active area of condensedmatter physics. The purpose of this chapter is to give a comprehensive and balanced account of both experimental and theoretical advances, focusing on the most important physics and, as much as possible, giving an intuitive picture of the new phenomena that have been observed and explained. I shall attempt to give an organized presentation of a large body of work that is dispersed in the literature. However, because of the sheer volume of the publications in this active field, rather than giving a shallow review of many articles, I prefer concentrating on the most relevant and fundamental work. In order to reach out to the condensed-matter physics community that is unaware of the recent advances, I will try to introduce as pedagogically as

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possible the basic concepts of nonlinear optical spectroscopy. For the specialists in atomic and molecular optics, I will present as progressively as possible some many-body concepts. Good interpretations of experiment often require sophisticated computational treatments in which one can lose track of the underlying physics. It turns out that it is possible to develop an intuitive model directly related to the correct theory that captures the essential physics, although it misses some details. The model is deduced from the general kinetic equations by averaging over band or excitonic indices to define an “effective” interband polarization that obeys a nonlinear Schrodinger equation. Each term of this equation has a simple and meaningful interpretation. I call it, and its generalizations, the eflectiue polarization model. I shall use it every time an intuitive explanation can clarify the interpretation of experiments, keeping in mind that a true comparison with theory can only be made through the full numerical treatment. This chapter is organized as follows: Section 11 discusses the nature and properties of the electronic elementary excitations of semiconductors near the fundamental band gap. Section 111 examines the time scales of the scattering processes and their basic descriptions. Section IV reviews the nonlinear optical effect in which only virtual electron-hole pairs are excited and uses this example as an introduction to the interpretation and description of the physics governing the coherent regime in condensed matter. Having introduced the main ideas of manybody interactions in semiconductor optics, the chapter turns to the processes involving excitation of real electron-hole pairs. Section V covers the fundamentals of processes that require accounting for two-particle correlations, and some applications to electronic states in heterostructures are described in Section VI. Section VII discusses experiment and theory of processes involving four-particle correlations. Section VIlI is devoted to the very early time regime where memory effects and non-Markovian dynamics dominate. Finally, Section IX gives some concluding remarks and lists a number of relevant topics that are not covered in this chapter because of space or because they are somehow less general.

11. Near-Band-Gap Excitations

The semiconductors of interest in this review are direct-gap zinc blende materials. The single particle states are well described by the effective mass approximation (EMA). Close to the point, the center of the Brillouin zone, four spin-degenerate bands contribute to the optical transitions. The conduction band, which originates from s orbitals, has the J = 1/2 symmetry

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and is spin-degenerate, J , = f1/2. The upper valence band originates from p orbitals and has a J = 3/2 character. It is split into the J , = f3/2 heavy-hole (hh) and the J , = & 1/2 light-hole (Ih) subbands which, in bulk materials, are degenerate at r but separate for k # 0 owing to their different curvature. The so-called spin-orbit split-off band, with J = 1/2, J , = & 1/2 character, is considerably lower in energy. The optical transitions induced by a* photons, which are in pure spin states, correspond to the AJ, = f 1 selection rule; they are sketched in Fig. 1. The sensitivity of the transitions to the photon polarization is often exploited in nonlinear optics to identify some pathways. In heterostructures and/or in a magnetic field, the dimensionality of the electmnic states is reduced by a potential that confines the particles in one or more directions (Bastard, 1988). Since this confinement depends on the mass, the hh-lh degeneracy is lifted, giving separate transitions to the conduction band. The particles are still free to move either in a plane (two-dimensional case) or along one axis (one-dimensional case). The band dispersion in the directions of free motion can be highly nonparabolic [18]. The degeneracy between the hh and the Ih bands also can be lifted if a stress is applied to the sample, thus reducing the crystal symmetry. In the description of optical properties of semiconductors, it is customary to introduce the joint density of states (DOS), . 9 N Ifor ( ~the ) transitions at

11/2,1/2>

w2, -1/2>

11/2y1/2>

11/2y-1/2>

FIG. 1. Sketch of the energy levels and selection rules for optical transition near the band gap of zinc blende and wurtzite direct-band-gapsemiconductors.

hw between the noninteracting particles of a valence band with energy ) a conduction band with energy dispersion ~ , ( k ) The . dispersion ~ , , ( k and general relation giving the density of states 9 ( w ) in terms of the Green’s function G(k, k‘, o)is r 9(w)=9 m

1

1

G(k, k’,w ) lk,k’

where

and the 4,(k)are the e-h eigen states of the system. In the simple case of noninteracting particles with parabolic band dispersion, these are plane waves. Because the bands form quasi-continua, the sum can be transformed into an integral, Ck+ fdkd, where the index d = 3, 2, 1 specifies the dimensionality, 3D, 2D, and lD, of the sample,

where E, is the band gap, m is the effective mass, A, = 1, 2 x and 4n, and O(ho/E, - 1) is the Heaviside step function. For lD, 9 ~ ~ ( has w ) a singularity at the band edge, 9#)(0) cc [ ~ , ( k ) q,(k) - h ~ ] - ” In ~ . 2D, %jff(w) is a simple step function, and in 3D, it has the usual 9,IvJ)(w)cx [c,(k) - ~ , ( k ) ho]“2profile. For OD, the 9:lvof(w)is simply a series of &functions. Although one can use these DOS for discussing single-particle effects, for example, as starting points for describing transport properties, this approach is very misleading in optics. It gives the impression that the Coulomb interaction is some kind of a refinement that is added a posteriori to improve an already pretty good description. This is completely wrong. The Coulomb interaction causes “zero order” effects that determine the very value of the optical band gap and simultaneously affect all optical transitions (Hybertsen and Louie, 1985; 1988; Louie, 1998). When such a transition occurs, the interaction of the particle promoted into the conduction band with those left in the valence bands is an integral part of the process. This “final state interaction” has profound effects on all optical properties. To introduce the most salient features, let us consider the simple twoparabolic-band model of a semiconductor. The ground state, full valence band and empty conduction band, is denoted 10).The simplest excited state, or exciron, has one electron removed from the valence band and one

3 Omicrr~PROCESSES IN SEMICONDUCTORS

181

put in the conduction band. Rather than using the creation and destuction operators in the conduction and valence bands, ti, tk,$1 and O,, it is sometimes more convenient to work in the electron-hole representation, + h - k and O k 4 f ; t k , with (21, &} = {hi, hk,} = dkkt,and ?i + 21, t k + t k , all other {. ,.>E 0. The operator creating an exciton with an electron at k, and a hole at kh is A

where the amplitude 4(ke, k,,) satisfies the k-space Schrodinger equation for an e-h pair interacting via the Coulomb potential. In the case of a photogenerated exciton, one can neglect the momentum of the photon so that the center of mass momentum is K = 0, and k, = - k, = k. Then the relative motion wavefunction satisfies the k-space Wannier equation:

4=(k)- C V ( k - k')4a(k')= 0

(5)

where the Greek index a labels the internal motion quantum number and runs over the bound and unbound states. Equation ( 5 ) is just the Fourier transform of the r-space hydrogen problem. It appears naturally in crystals where, because of Bloch's theorem, it is usual to work in k-space. It shows explicitly that in k-space the Coulomb interaction ZkrV ( k - k')$a(k') couples states at different k. The solutions of Eq. ( 5 ) forms a complete basis and satisfy the closure relation

Using these eigen-functions in Eqs. (1) and (2) gives 9+), the DOS for the Coulomb interacting and optically active pair states:

where r = re- r, is the e-h relative coordinate. For noninteracting particles where V ( k ) + 0, the Cp,(k) are plane waves with 4a(r = 0) = 1, and one = xkd[&,(k) - &,(k) - ho]. recovers the result of Eq. (3), gNI(o) When the Coulomb interaction is accounted for, only the s-like pair states with &(r = 0)ZO contribute to the optical transitions. Equation (3)

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D. S . CHEMLA

includes the bound and unbound states and therefore accounts for the contribution of the resonances as well as the Sommerfeld enhanced continuum. It implies the famous Elliott formula for the linear absorption [21], which for the 3D case is written as

with

measures the photon excess energy in units of the exciton Rydberg Ry. Here pcEis the interband transition dipole matrix element, which can be consider-

ed as independent of k with an excellent accuracy; a, is the exciton Bohr radius, and no is the background refractive index. Similar expressions for 2D and 1D can be found in many textbooks. One significant implication of the Elliot formula is that the absorption remains well above the noninteracting particle result aNla @(AQ)&Q even far away from the gap (for AR z 50, a J Dis still about 10% above anr).It turns out that an analytical expression for the real part of the dielectric function, and hence for the refractive index, can be derived as well (Tanguy, 1995). In Fig. 2 the solid lines give the experimental low-temperature (T = 5 K) absorption and refractive index spectra of a high-quality 1-pm-thick GaAs sample. The sample is antireflection coated on each side and glued on a sapphire substrate. Stress due to different thermal expansion lifts the Ih-hh degeneracy. One can distinguish clearly in the a ( o ) spectrum the two exciton peaks and the onset of the continuum. The small bump at the threshold of the latter is likely due to the 2s resonance. All these features have a “dispersive” correspondent in the refractive index spectra. On the scale of Fig. 2, the noninteracting particle absorption would barely be visible. The dotted lines show a fit to the analytical expressions (Elliott, 1957; Tanguy, 1995). Although the main features are reproduced accurately, some details that are smoothed out in the experimental curves are more apparent on the theoretical ones. Strictly speaking, as soon as a second pair is created, one should account for the interaction between the four particles. Intuitively, however, one expects that as long as the distance between the two pairs is much larger than a,, the preceding results should not be affected too much. In order to be more quantitative, let us compute the commutation rule of the exciton operator. One finds, as a function of the number operators for electrons,

3 OPTICAL PROCESSES IN SEMICONDUCTORS

6

I pm GaAs T=5K

c

'

-0.2 1.505

183

1.510

I

I

1.515

1.520

Energy (ev) FIG. 2. Solid lines: Experimental spectra of the absorption coefficient and refractive index of GaAs at the fundamental absorption edge. The I-pm sample is at T = 1.6K;it is glued on a sapphire substrate, and the thermally induced stress has shifted the heavy-hole-light-hole degeneracy. Dotted lines: Fits to the analytical formulae discussed in the text.

in which we recognize a form of the Pauli principle. We need the expectation values of A&) and A,(k) in the exciton state IX,) = giglfa). It is easy to

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D. S. CHEMLA

show that

so that

Hence, from the point of view of quantum statistics, creating one exciton is equivalent to creating a “very special” distribution of electrons and holes: l+,(k)I2. This result implies that the generation of excitons prevents the subsequent generation of other excitons, a behavior of Fermions and not of Bosons. Quite often it is found in the literature that excitons behave as Bosons because they are made of an electron and a hole with opposite spin. This is wrong; at most, one can consider that excitons are nonideal “composite” Bosons with underlying Fermi statistics reflecting that of their constituents. Creating an exciton uses some states of the pool of Fermionic states of the crystal and produces Pauli blocking (PB). This effect is called excitonic phase space filling (PSF) in the literature. The magnitude of the exciton Pauli blocking can be estimated by noting that for the bound states 4,(k) ‘v 0 for k >> a; for example, +&) = 8&xa2)/[1 ( k ~ ~ There) ~ ] ~ . fore, returning in r-space to express the result intuitively, once an exciton is created, a small volume of semiconductor -a: can no longer sustain other excitons. Since the center of mass momentum vanishes, K = 0, the “location” of this “excluded volume” is not known; neverthless, the net effect of creating an exciton is to decrease the absorption of the sample. The exact value of the “excluded volume” has to be calculated from more precise manybody theory considerations (Schmitt-Rink et al., 1989). The approximately Bosonic behavior of excitons can only be observed in the low-density regime N x u(: << 1. Let us note for future reference that for the same reasons, a plasma with e and h distributions, n,(k) and n,(k), also causes a PSF due to the overlap between these distributions and +Jk). As the density increases, excitons begin to interact, and one expects, in analogy with the hydrogen problem, that two pairs would bind to form an excitonic molecule or biexciton X,. Theoretical considerations show that the singlet state is indeed bound for all values of the electron-hole mass ratio, whereas the triplet state is unbound, i.e., X-X.Biexcitons can be created directly by a two-photon transition or via relaxation and binding in an exciton population. Therefore, they are active in coherent processes involving two or more photons. The bound singlet state X , requires two photons of opposite polarization to be excited. This is achieved either by

’,

+

3 OPTICAL PROCESSES I N SEMICONDUCTORS

185

two distinct laser beams, one o+ and one o-, or by use of linearly polarized beams. For future reference, let us note that the four polarization configurations that are two-photon active are often called cocircular (o*/o*), countercircular (o*/cT), linear parallel (11-polarized), and linear perpendiculur(l-polarized). Bound biexcitons have been observed in bulk large-gap semiconductors, II-VI and I-VII, but not directly in III-V semiconductors because, in these materials, the binding energy is very small, < 1 meV. In quantum confined systems, heterostructures and/or under large magnetic fields, the biexciton binding energy is enhanced, and as discussed in Section VII, they play an important role in nonlinear coherent processes. In quantum confined systems, a potential, either modulation of the band gap in heterostructures or cyclotron energy in magnetic fields, quantizes the e and h motion in some directions. Let L j be the effective confinement length in directions Zj. Then the corresponding components of the momentum can only take quantized values k j x nj(2n/Lj ) , and the particles acquire confinement energies E , k ( n j ) % C,,(hkj)2/(2me,h).Usually, because of the symmetry of the envelope functions, the strongest optical transitions occur between e and h subbands with the same index nj. They have onsets at ho = E, + Ee(nj) + Eh(nj), followed by continua associated with the free motion in the unconfined directions. Therefore, one expects to see an excitonic structure at each one of these thresholds. This is the case with, however, an interesting twist for the high-energy transitions. Their excitons overlap in energy with the lower-energy continua and are coupled to them by the Coulomb interaction (Glutsch et al., 1994; 1995). The true eigen are superpositions of the (nj) excitons and the (nl< nj) states, I$(&) , continua. The is the recipe for a quantum interference that produces highly asymmetric “Fano” resonances (Fano, 1961). Let us consider for simplicity the case of a single discrete state, IX),coupled to a featureless continuum, JC(&)),and let V be the coupling constant. Then, even in the absence of any other dissipation mechanism, the Fano resonances acquire a linewidth = nV2 and a transition dipole moment

where A o = ( E - hw)/I‘,and q = Ipxg/pce12/7cVis i..e Fano parameter that measures the ratio of dipole moment of the discrete state and the continuum. The transition probability, a (p+(e)B12,vanishes for A o = q and has a maximum at A o = - q - ’ , thus giving the very asymmetric profile to the absorption spectrum. Fano resonances have been observed in heterostructures and in bulk semiconductors in magnetic fields (Glutsch et al., 1994;

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D. S. CHEMLA

1995). They do not correspond to bound states and should be considered as “structured continua.” An example of Lorentian excitons and Fano resonances observed in the absorption spectrum of GaAs at a magnetic field of €3 = 12T for the two polarizations a’ and CJ- is shown in Fig. 3. The case where a large density of carriers is already present in the sample is also very interesting. In that case, the common wisdom would say that the Coulomb interaction is screened and that one should recover the noninteracting particle results. Again, this is far from being the case. To illustrate what occurs in the presence of carriers, let us consider the optical transitions in modulation-doped quantum well (QW) or quantum wire structures. In these heterostructures, dopants are introduced in the large-gap material during the growth. The dopant nuclei remain locked in that material, but the carriers move to the lowest gap in the structure, where they form extremely high mobility electron or hole gases. To discuss a specific situation, let us assume an n-doped Q W structure where electrons form a 2D electron gas and therefore fill the conduction band up to a Fermi energy E,. Because of final state occupation in the noninteracting particle picture, one expects that the transitions vary as a 9#)(0) x [l - n,(w)].Therefore,

1.5

1.52

1.54

Energy (eV)

FIG. 3. Absorption of a I-pm GaAs sample in a B = IOT magnetic field for o+ and 4polarizations. Regular Lorentzjan exciton resonances are Seen at the lowest edge; at the higher Landau edges strong Fano resonances are observed.

3 OPTICAL PROCESSES IN SEMICONDUCTORS

187

at low temperature, the absorption profile should vanish around E,, have a threshold around E,, and recover its usual profile above E F , i.e., exhibit the Burstein shift. This is not at all what is observed; in fact, close to E, the absorption exhibits a strong spectral peak called a Fermi edge singularity (FES) (Skolnick et al., 1987; Livescu et al., 1988). An example of an FES is shown in Fig. 4, where the absorption spectra of an undoped and a modulation-doped GaAs/AlGaAs Q W structure are compared (Brener et al., 1995). At low temperature, the FES produces a spectral feature as strong as that of the excitons! The FES originates from Coulomb-mediated static and dynamic manybody effects. The physics behind the FES is that in an absorption process the photoexcited hole interacts not only with its photoexcited electron companion but also with the electrons of the entire Fermi sea (Mahan, 1967; 1988). Furthermore, the dynamic Coulomb effects result in a complicated reaction of the Fermi sea at the sudden appearance of the photoexcited hole. Electrons inside the Fermi sea, with momentum qe < k,, interact with the photohole and scatter to states of momentum pe > k,, while the photohole recoils from k h to k;, = k, qe - pe. It is necessary to introduce this heavy notation to distinguish between the e and h photogenerated in the interband optical transitions and those created in the conduction band by intraband transitions that are due to the dynamic readjustment of the Fermi sea. It turns out that this readjustment cannot be described by a perturbative approach. The first-order perturbation amplitude for the scattering described above is a VJAE, where V, is the screened e-h interaction and AE = ~ , ( p , )- c,(qe) (m,/mh)[&,(k’h)- &,(kh)].Close to E,, AE can be vanishingly small, and the number of scattered particles can become very large. The combination of an infinitely large number of processes of vanishing energy leads to a breakdown of the pertubation expansion! A number of approximations, very often used for treating manybody effects, “rigid Fermi sea approximation,” or energy-independent Coulomb interaction, fail completely and lead to unphysical results. In fact, the dynamic readjustment of the Fermi sea gives an energy-dependent e-h interaction. Recently, a nonperturbative approach was developed for the case of a hole with an infinite mass. It has been shown that the final-state wavefunction is a coherent superposition of all the Fermi sea excited states (Perakis and Chang, 1991a; 1991b; 1993). In the case where the h has an infinite mass, the absorption profile exhibits a power law divergence (Nozieres and de Dominicis, 1969)

+

+

I

r

IS

where RT is the threshold transition, to is a typical conduction bandwidth, and B is a function of the screening charges in the immediate vicinity of the

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D. S. CHEMLA

1.6

1.5

1.7

1.8

Energy (eV)

FIG. 4. Comparison of the absorption coefficient of(a) a modulation doped (solid line) and tb) an undoped (dotted line) GaAs quantum well structure at T = 8K. The Fermi edge singularity at the edge of the modulation-doped sample spectrum is as pronounced as the regular excitons in the spectrum of the undoped sample.

hole (Hopfield, 1969). Finite h mass raises the difficult question of the h-recoil. A more complete treatment shows that this produces two thresholds, one at E , (1 + m,/rn,)EF for the vertical transitions, whereas the indirect transitions start at E, EF, with creation of conduction band e-h pairs and plasmons for conservation of momentum. The singularity is smeared over an energy range on the order of the hole band width and is therefore changed into a peak (Gavoret er a/., 1969; Uenoyama and Sham, 1990). This leads to the broad spectral feature seen in Fig. 4. The theory is not yet finalized, due to formidable difficulties, such as the nonlocal e-h interaction that the valence band dispersion and the conservation of the total momentum impose. This concludes the brief review of the dominant elementary excitations that are involved in near-band-gap optical processes. The next section considers the time scales of their dynamics.

-

+

-

+

111. Time Scales and Dynamic Trends

When discussing coherent processes, it is important to compare the time scales of the optical excitation with those of the “fast” and “slow” electronic

3 OPTICAL h o c ~ s s IN ~ sSEMICONDUCTORS

189

degrees of freedom that interact with it. For the light field, E(t) = &(t)exp( - iot) + c.c., one can distinguish the duration of the optical cycle T = 274~0of the carrier wave from the characteristics of the pulse envelope &(t). For excitation close to the band gap of common semiconductors, the period of the optical cycles T x 27rh/E, is in the 2-fs -+ 3-fs range. Modern ultrashort pulsed lasers deliver 5-fs -+ l-ps pulses. It is customary to assume that the optical field drives interband e-h charge fluctuations that can follow the carrier wave and to make the so-called slow varying envelope approximation for discussing light-semiconductor interactions. This separates the fast variations of the material parameters at frequency w from the slow variations that follows &(t). For example, in the case of the polarization, one assumes the ansatz P(t) = P(t) exp(-iwt) + C.C. The problem is further simplified by the rotating wave approximation (RWA) that retains only the resonant or quasi-resonant terms of the equations of motion. Although these approximations may be legitimate for the longer 100-fs -+ l-ps pulses, for the shorter ones, containing less than ten cycles, they may be problematic. The general theoretical approach for describing optical processes in semiconductors is to try to solve the kinetic equations of the density matrix. For the off-diagonal elements and the diagonal elements, i.e., the polarization amplitudes pk and the occupation numbers ne(k) and nh(k), respectively, the kinetic equations take the form

and

The coherent parts of Eqs. (13) and (14) are derived from the Heisenberg equations and are discussed in Sections IV and V. This section analyzes the scattering terms. The nature of the electronic species that are excited and the time scales over which they can maintain well-defined phases depend on hw relative to the band gap. It is clear from the preceding section that one can roughly distinguish three excitation regimes: well below the lowest energy resonances, around the resonances, and well above them. In the first case, the excitation energy hw falls in the transparency range of the material. During excitation, the quantum state of the sample can be described as a linear superposition of excited states; i.e., only “virtual”

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D. S. CHEMLA

excitations are generated. If we define a typical detuning as Aw = Re,, - w, the uncertainty principle tells us that the “lifetime” of the virtual excitation is z z (Am)-$. During that lifetime, the virtual excitations possess all the properties of real ones; however, they do not participate in the dissipation processes (or they would become real), and they disappear over a time on the order of T once the excitation ceases. This is a true coherent regime where the virtual excitations are completely driven by the light field, and they are described by Eqs. (13) and (14) with the last term set equal to zero. The corresponding nonlinearities are small, but they are extremely fast. Both properties stem from the fact that Aw is large. This case is covered in Section IV.

In the two other cases, the photons generate real e-h pairs, whose quantum mechanical phase is well defined only initially. These e-h pairs can participate in coherent processes over a time scale on the order of a dephasing time T2 that is determined by the scattering rates Ti;that is T, = (Ziri)-’. It is clear that these differ widely depending on whether the e-h pairs form bound states and/or are strongly correlated or have a significant relative kinetic energy. Relaxation affects more strongly the high-energy carriers, but even for these it is not an instantaneous process. In the few ferntoseconds following excitation, manybody interactions start to take place, destroying the e-h quantum mechanical phase coherence and creating “real carriers.” A semiconductor, as any quantum mechanical system, is described by a Schrodinger equation that is local in time. Therefore, strictly speaking, knowledge of the Hamiltonian and the present state is enough to determine the future evolution. In practice, however, the semiconductor is such a complex manybody system that this ideal program cannot be achieved. It is customary to divide the full system into a subsystem that is analyzed and a thermal “reservoir” of all the other degrees of freedom on which we have only partial information, e.g., respectively, the interband transitions and the phonons and other carriers. Then one can distinguish two stages in the dynamics of the subsystem: the long-term behavior occurring when scattering processes have randomized the degrees of freedom, which is well described by Boltzman kinetics, and the behavior at very early time where it is necessary to follow up the uncompleted scattering processes in the subsystem-reservoir interaction. This stage of the dynamics is highly nonclassic; the uncertainty principle tells us that the energy of each e-h pair is not well defined; populations and polarizations, i.e., diagonal and off-diagonal elements of the density matrix, are strongly coupled; and the lattice and the photoplasma are just starting to react. In this regime, the dynamics of the polarization and k-dependent occupation numbers must be described by Quantum kinetics, with non-Markovian statistics and memory structures. A comprehensive review of the current

191

3 OPTICAL PROCESSES IN SEMICONDUCTORS

theory (Haug and Jauho, 1996) of that early stage of relaxation is beyond the scope of this chapter. However, it is useful to consider a very simple model for introducing the main concepts and discussing the essential physics (Haug and Koch, 1993). In this model, the subsystem is a harmonic oscillator i coupled to a bath of other harmonic oscillators A^,. The Hamiltonian is

The Heisenberg equations of motions for 2 and d --2= dt

-ihwk-iCg,A,

,.

and

A, are

d-.

-A,= dt

A

-ihf2,Av-igv2

The last equation can be integrated formally, &t) = - ig, JL d t ‘ i ( t ’ ) exp[ - iR,(t - t‘)], assuming A,(- co) = 0, and put back in the first one, which takes the form d

-i dt

=

- ihw?

+

s1

dt’r(t

- t‘)?(t’)

m

where we have introduced the memory kernel r(t - t ’ ) = ihZ,g, exp[ -iRv(t - t ’ ) ] , characterizing how long the subsystem remembers its past. Subsystems with an ultrashort memory, r(t - t’) = y6(t - t’), are called Markovian, and their relaxation can be described by a single dephasing time T2 = y-l. In this oversimplified model, the Markovian regime is obtained by making a number of simplifying approximations: (1) the reservoir is assumed to have a dense and featureless continuous spectum with density of states .9(w), (2) the coupling is assumed to be weak, g , << R,, so that one can develop the equations of motion to first order in g,, which is furthermore taken as a constant g , !z Q, and more important, (3) all the transient effects are neglected by extending the integrals to t + coy thus introducing energy-conserving &functions, limt-r {g, f‘- d t ’ i ( t ’ ) exp[ - iR, (t - t ’ ) ] ) + in@?((t)s(R,- w ) giving y x 2nhg29(w). This procedure is equivalent to calculating the scattering rates using Fermi’s golden rule. This almost trivial example illustrates how memory effects manifest themselves in general: At time t’ some dynamic variable of the subsystem interacts with the reservoir and drives it; the degrees of freedom of the reservoir then evolve according to their own eigen energies; and in turn, at a later time t the reservoir interacts with the subsystem, carrying information about the dynamic variables of the subsystem at time t’ as well as its

+

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D. S. CHEMLA

own dynamics between t‘ and t. Depending on the nature of the subsystem and of the reservoir, the kinetic equations can be more or less complicated to write and solve, but the behavior of the simple example just sketched is generic, In general, because of the very large number of degrees of freedom of the reservoir, interferences are the rule between t and t‘, and information is lost (from the point of view of the subsystem). This “scrambling” is not instantaneous, and often it is possible to define a correlation time T~ that characterizes the memory kernel; that is, T(r) # 0 for t < T~ and r(t)+ 0 for r >> 5,. For t - t’ < rc, the scattering integrals must be calculated explicitly. It is clear that they have an oscillatory behavior with frequencies related to the natural frequencies of the reservoir Qv. These oscillations translate the wave nature of the elementary excitations and the finite duration of their “collisions.” Let us note that because t is small, energy conservation is not strict in the scattering events but is limited by the time/energy uncertainty relations. I t is also worth noting that the time domain memory kernel corresponds by Fourier transform 9.5. to energy-dependent scattering rates, 9.3. [jr(t - t ‘ ) i ( t ’ ) d t ] + T(w)%w), in the frequency domain. To apply the line of thoughts just sketched to realistic situations, one has to turn to much more powerful theoretical techniques and more sophisticated treatments (Binder and Koch, 1995; Kuznetsov, 1991a; 1991b; El Sayed and Haug, 1992; El Sayed et al., 1992; 1994; 1994-1; Hartmann and Schafer, 1992; Thoi and Haug, 1993). The main dephasing mechanisms in semiconductors are the electronphonon and the electron-electron interactions. Most of the direct-gap semiconductors are at least partially ionic, and the dominant coupling with the phonons is due to the Frolich LO-phonon-carrier interaction. The time scale of the initial non-Markovian regime can be estimated by noting that the time it takes a lattice or a plasma to react to a perturbation is on the order of one period of the natural oscillation (El Sayed et al., 1994-1). For example, in the case of GaAs, the LO-phonon period is TLo= 2 4 flto = 115fs, and for a plasma of density of nrh = 10’8cm-3 in the same materal Tpl= 27c/Rp1z 1OOfs. The non-Markovian regime in phonon scattering of photocarriers generated by ultrashort pulses was analyzed recently revealing some features specific to the early time domain (Kuznetsov 1991a; 1991b). These carriers are not classic; each one is spread over a large portion of k-space by the uncertainty principle. This spread must be accounted for in each scattering event. At an early stage of the relaxation, pn and n,,,(k) are coupled and cannot be analyzed independently. This “polarization scattering” is discussed in Section IV. Collisions are “incomplete” in the sense that carriers do not lose or gain exactly one phonon because their energy is not well defined. Consequently, the collision integrals exhibit a “memory behavior”;

3

OF’TICAL PROCESSES IN SEMICONDUCTORS

193

for example, in the case of electrons, they have the following structure:

+ (NLO(4,

t’)

+ l)cosCS+(t - f)l)

(17)

with <* = [ce(IC) - ~ , ( k- q) & QLo], plus terms that account for polarization scattering (Kuznetsov 1991a; 1991b). Here N L O is the LO-phonon occupation number. Note the oscillatory functions of time under the integral; they describe the “incompleted scattering events. It is interesting to note that because the dispersion of LO-phonons is negligible, these lattice vibrations form a “single-mode reservoir” whose frequency explicitly appears in Eq. (17). For the reasons evoked earlier, in the Markovian limit t + 00, the oscillatory functions of Eq. (17) transform into energy-conserving 6 functions, and the collision integrals take a more familiar form:

ata nelseatt

0~ 2

4 t -)[I - ne(k, t)lne(k

- 4, t)NLo(q, t )

+ 2~6(<+)[1- n e ( k t)Ine(k - 4, t)CNLo(q, t ) + 11

(18)

with phonon emission and absorption terms multiplying the Pauli blocking factors. In particular, this implies that in the non-Markovian regime the rates of scattering-in and scattering-out of one state are modified as compared with the Boltzmann case. In the latter they are two Lorentzians (broadened 6-functions) with opposite sign and separately by haLO. In the non-Markovian case (Fig. 5), the region of k-space involved in the scattering is only defined with an accuracy h/(t - r’); i.e., a much wider region of phase space is accessible than for the exchange of exactly one phonon, resulting in a very broad scattering-in rate and a slightly reduced scattering-out rate. In the long time regime, the steady-state scattering rate is given by the Fermi golden rule (Shah, 1996):

+ (NLO+ 1)sinh-’ r;iL: -I]] where

D.S. CHEMLA

194

-1 2

Excess energy FIG. 5. Phonon scattering-in and scattering-out rates calculated with non-Markovian and Boltzmann kinetics theories according to Kuznetsov (1991). In the non-Markovian case, the region of phase-space accessibility is much broader than in the Boltmann case.

Here E , and co are the material optical and static dielectric constants, respectively, and m(e,h)is the e- or h-effective mass. There is evidently a threshold for the emission of LO-phonons at energy Ee,h(k)x E , + hRLo and no threshold for the absorption. The time scale for the establishment of the steady-state regime is given by rbLo) z (125fs)-' in the case of a moderately ionic compound like GaAs. I should mention that the deformation potential scattering couples the lattice to carriers in all materials, purely covalent or ionic. Acoustic phonon interactions result in much slower scattering rates. The phonon scattering rates for excitons usually are calculated independently for the electron and the hole with a form factor that accounts for their relative motion (Ridley, 1982). The steady-state dephasing rate of excitons can be estimated from their spectral linewidth. A t !ow temperature, it is limited by the defects in the sample. In high-quality materials, the exciton dephasing time is in the range 0.5 ps < T2 < 10 ps. As the temperature increases, phonon scattering shortens T2.It is found experimentally (Chemla et al., 1984), and theoretically substantiated, that the exciton linewidth varies approximately linearly with the density of

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thermal LO-phonons, T x x To + Tph x NLo(T), resulting in dephasing times T2z lOOfs - 200fs at room temperature. Dephasing through carrier-carrier scattering is much more difficult to describe. In the case of interaction with phonons, the lattice is the thermal reservoir, and the electronic degrees of freedom are distinct from those of the reservoir. Carrier-carrier scattering proceeds from the Coulomb interaction, the very force that determines the nature of the elementary excitations. In a sense the “subsystem” is its own “reservoir.” The screened Coulomb potential K(r, t ) itself is a retarded function with its own kinetics. In particular, screening is not instantaneous; it builds up over a time scale Tp, and for shorter times is essentially unscreened, that is, K(r, t x 0)-,V ( r ) (El Sayed et al., 1994-1). Carrier-carrier scattering is often described in terms of the semiclassical Boltzmann kinetics (Goodnick and Lugli, 1988; Collet, 1993). On the very short time scale, this formalism is plagued by a number of problems related to the fact that it conserves exactly the kinetic energy, is local in time, and thus violates the uncertainty principle. On that time scale, carrier-carrier scattering has to be described by Coulomb quantum kinetics (CQK). As for phonons, CQK involves collective effects, and a number of features discussed earlier in the case of phonons remain valid. For example, the carrier energy is not well defined, and this broadening can be larger than the typical energy transfer in two-particle collisions, which, therefore, are “incomplete” (El Sayed et af., 1994). As opposed to LOphonons, however, an electron gas has a gapless spectrum that cannot be characterized by well-defined and discrete frequencies. Therefore, in general, carrier-carrier scattering tends to broaden significantly and quickly any distribution of electrons or holes. CQK expresses scattering in terms of transient scattering integrals with a memory structure. These have exactly the same structure for e and h (Haug and Jauho, 1996):

-

a at

i

x {ni(k, t‘)ni(k’, t’)[1 - ni(k - 4, t‘)][1 - nj(k

+ q, t’)]

Here i, j d e , h, V ( 4 ) is the unscreened Coulomb potential, and A&= Eio)(k) + ~$O)(o‘(k’) - &io)(k - q ) - cf0)(k‘ + 4), where the &Io)(k) are the bare energies; see Section IV. The transient scattering integrals satisfy the Pauli principle, as seen from the scattering-in and scattering-out terms, and conserve the number of carriers and the total polarization. Conservation of the kinetic energy, however, is not strict, as expressed by the terms a

196

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cos A4t - r'). As a result, a carrier distribution injected in the band spreads very quickly over a portion of k-space much larger than predicted by Boltzmann kinetics. The associated broadening tends to attenuate the visibility of the phonon sidebands. Since, in the early time n k x IpkIz 2 1E12, the initial scattering rates scale as the excitation intensity. An example of the time evolution of a distribution of electrons generated in 20fs calculated with CQK formalism is shown in Fig. 6, One can clearly see a considerable broadening of the distribution, even during carrier injection. Current CQK calculations, however, have an infinite memory depth; the transient scattering integrals can have an unstable evolution at long times, leading to overshoots in the carrier distributions. These theories therefore are only reliable for describing the early-time behavior, whereas Boltzmann kinetics theory is valid in the other limit of long times. Making the link between the two time scales still represents a formidable theoretical challenge. It is worth noting that a very different description of early-time dephasing has been proposed (Gurevich et al., 1990). Because, immediately on creation,

FIG. 6 . Time-momentum evolution of a population of electrons generated in 20fs in the conduction band of GaAs calculated with quantum kinetic theory. (Courtesy of Prof. K. El Sayed.)

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a photoplasma is still immobile, one can consider that its effect is to create a random potential that is associated with an inhomogeneous broadening and thus leads to a dephasing mechanism. The phase decay is obtained by averaging over the disordered potential, giving, in bulk materials, a nonexponential decay for the polarization Ipk(t)l a e ~ p ( t / z )with ~ z-3 a Rj?(nai)(kao). The cubic time dependence is related to the dimensionality of the system; in 2D it becomes a quadratic dependence. By analogy with atomic physics, dephasing due to interaction between carriers is sometimes described as a source of “collisional broadening,” and the effect of electrons and/or holes on a linewidth is written as y = yo + yene + yhn, +.... Although intuitively attractive, this generalization can be misleading, especially in the context of coherent processes. Putting, without precaution, this expression in Eq. (13) can lead to severe contradictions. To see this more clearly, let us consider the case of a polarization component pk = (i?khk). The product ypk = (yAo yene + y,nh)p, contains terms of the form (i?t,i?k*)(f?k&) and ( ) ; , $ h k ~ ) ( i $ i k ) , which, in fact, originate from the uncontrolled factorization of the four-particle operator products i?$?,.&,h, and fi&i?kfik. The coherence of these quantities must be accounted for by the equation of motion of the corresponding four-particle correlation functions, which are coupled among themselves and with that of the density matrix. Since e-h pairs, just on generation, have a well-defined phase, the distinction between coherent and incoherent carrier-carrier interactions is a nontrivial matter that must be considered very carefully for each case; see Section VII. For example, when carriers form a Fermi sea in a sample, the Laundau theory of Fermi liquids predicts that the lifetime of quasi-particles with excess energy 88 varies as z a (E,/&)’ and therefore becomes infinite on the Fermi surface. Evidently, this has important consequences for all coherent optical processes. Neverthless, it is true in general that the presence of a large number of carriers, photogenerated or not, reduces significantly the period of phase coherence. In summary, following the excitation of a semiconductor by an ultrashort laser pulse, one can distinguish the following time sequences: Initially, during the first few optical cycles, e-h pairs oscillate coherently between the valence and conduction bands; then, typically in a few tens of femtoseconds, manybody interactions start to destroy the phase coherence. As e, h, and nuclei start to move, a number of processes begin to be “turned on,” the Coulomb potential, which is initially bare, starts to be screened (El Sayed et al., 1994), and the lattice begins to react to the appearance of charges (Kuznetsov 1991a; 1991b). During this transient period, the non-Markovian dynamics are described by quantum kinetics. Although the carriers start to relax, their distribution cannot be described by Fermi functions, and a “temperature” cannot be defined. As scattering processes become effective,

+

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the coherence continues to decay and is completely lost after several tens of femtoseconds when the quasi-particle scattering rates have reached a regime where the occupation numbers follow Fermi-Dirac statistics and the system can be described by Boltzmann kinetics. Early in this regime there is still no equilibrium among different carrier species or between the carriers and the lattice, but carrier-phonon and carrier-carrier scattering recovers its usual behavior. Eventually, the carriers equilibrate first among themselves and later with the lattice before finally recombining.

IV. A Purely Coherent Process Involving Only Virtual Electron-Hole Pairs: The Excitonic Optical Stark Effect A true coherent state of a semiconductor is realized when the material is excited well below its optical transition energy threshold by a coherent laser field. Then only virtual excitations are generated. They do not participate in dissipation, and their phase is determined by that of the laser. Such situations also can be realized in atomic systems, where it is well known that the laser field induces the so-called optical Stark effect (OSE), whereby the k-atomic level experiences a shift in energy:

where pk is the transition dipole moment, and A&@)= h(R,, - o)is the laser detuning (Feneuille, 1977). The Rabi frequency I/.$* El measures the atomEM-field coupling. An apparently similar effect was observed in semiconductor Q W structures excited in their transparency range. Ultrashort-pulse pump/probe experiments revealed, indeed, that the exciton resonances experience a shift that follows instantaneously the laser pulses (Fig. 7) with a magnitude inversely proportional to the detuning and proportional to the laser intensity, in agreement with Eq. (21) (Mysyrowicz et al., 1986; von Lehmen et af., 1986). In these experiments, the small changes in sample transmission seen by a weak probe laser and induced by a strong pump laser are measured. In the small signal regime, the differential transmission spectrum (DTS), A T / T = [T(I,) - T ( I p= O)]/T(lp= 0), reproduces faithfully the changes in the absorption spectrum of the sample c((o),since A T / T s - A a ( o ) x 1. A number of reports confirmed these early observations and provided more detailed information on the excitonic OSE (Tai et a/., 1987;Joffre er al., 1987; 1989; Knox et al., 1989; Chemla ef al., 1989). The most interesting aspects of the excitonic OSE, however, relate to its interpretation (Schmitt-Rink and Chemla, 1986; Schmitt-Rink et al., 1988).

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-7

-

FIG. 7. Excitonic optical Stark effect observed in ultrafast time-resolved pump/probe experiment. (Mysyrowicz et al., 1986). The pump pulse duration is IOOfs, and the sample is a IGnm GaAs quantum well structure. The pump spectrum shows where, below the resonances, the sample is excited. The absorption spectra for At = -2 ps, Ops, and + 1.2 ps are given by the solid line, the dotted line, and the dashed line, respectively.

Because of its importance as an introduction to the physics that governs light-semiconductor interactions, it is worth discussing the fundamental issues. Furthermore, the excitonic OSE provides us with a very natural way of introducing the theoretical basis for the coherent part of the kinetic equations, i.e., the first part of Eqs. (13) and (14). One approach for decreasing the excitonic OSE could be to start from the quasi-continuum of valence and conduction band states and apply to each transition at k the atomic picture. Considering for simplicity the twoparabolic-band model, the levels at the bottom of the conduction band and at the top of the valence band are repelled more than the higher levels because of their smaller detuning Ac(k). Hence the curvature of the band is modified by the laser field, and the effective masses are renormalized by the light! Since excitons are made of the (k Ia; ’) states, the “new” excitons would be constructed from these dressed single-particle states. This approach gives the priority to the single-particle-EM-field interaction over the

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Coulomb interaction between particles, but as shown in Section 11, this is not justified. The alternative approach is to treat the excitons as eigen states of the crystal and apply to them the atomic picture. In this case, the priority is given to the Coulomb interaction and not to the interaction with the EM-field. The charges, however, react to the total field, whatever its origin, the laser or other charge oscillations in the medium. This lengthy discussion is aimed at demonstrating that one must treat the Coulomb interaction and the interaction with the EM-field on the same footing. This is true for any light-semiconductor interaction and causes some theoretical difficulties. Let us apply this program to the two-parabolic-band model. The basic Hamiltonian of the electronic system is

where the first line describes the dispersion in the conduction and valence hands and the second line the intraband and interband Coulomb interaction. For the purely coherent excitonic OSE, we do not need to consider dissipation; i.e., we can put i $ f / d t l s c a t t E 0 in Eq. (13), and therefore, as a starting point we do not have to explicitly involve electron-phonon and carrier-carrier scattering. The light-matter interaction is treated in the semiclassic dipole approximation and the rotating wave approximation with the interaction Hamiltonian

a, = -c

[h.kE(t)c^:fik

+

&.kE(t)*v*::k]

(23)

k

The total Hamiltonian, PI,,= fie,+ H , , has no known solutions, and approximations are necessary at this point. The general approach for describing optical properties of a system is to determine the expectation value of operators such as the polarization P through the density matrix operator A by P = Tfliip)(Shen, 1984;Mukamel, 1995). In the two-parabolic-band model, the matrix elements of 6 are the expectation values of two-particle operators:

The normal procedure for getting the equation of motion of A is to write the to apply the Heisenberg equations for the two-particle operators using h,,,,

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fundamental anticommutation rules, and then to take the expectation values. Because fit,, contains four-particle operators, these appear in the two-particle operator equation of motion, so one has to write the Heisenberg equations for the four-particle operators. These in turn contain six-particle operators, Ieading to an infinite hierarchy of coupled equations. This behavior is a fundamental problem of manybody systems that can find many embodiments in various formalisms (Feynman diagrams, Green's functions, etc.); in all cases, when an exact solution is not available, one has to decide on a way to truncate this development. The simplest approach, beyond the independent-particle model, is to factorize the expectation values of the four-particle operators into products of expectation values of twoparticle operators, i.e., performing the random phase approximation. This is equivalent to the time-dependent Hartree-Fock treatment (Lindberg and Koch, 1988-1). At this HF/RPA level, correlation between four or more particles is neglected, and we will see in Section VI schemes for accounting for them. With all these approximations and considering only vertical transitions because the photon momentum is negligible, the density matrix breaks into 2 x 2 blocks:

It obeys the Liouville equation of motion, without the usual relaxation term, (a/at)iik(t)l

seatt

:

and the effective HF/RPA Hamiltonian

Equation (27) immediately shows that the Coulomb interaction renormalizes, at the same level, the energies and the coupling to the EM-field, as measured by the Rabi frequency:

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Equation (28) expresses a renormalization of the single-particle energies caused by the virtual populations; it is reminescent of the band-gap renormulization (BGR) that is often considered in a different context for incoherent electrically injected carriers or thermalized photocarriers. Coherent populations also cause a BGR, and since in that regime the polarizations have not dephased, the transient BGR (Eq. 28) is accompanied by a similar transient renormalization of the Rabi frequency (Eq. 29). This equation simply expresses that carriers experience the “total” field laser plus Coulomb coupling to other dipoles. The Liouville equation (Eq. 26) then becomes

These equations have two constants of motion:

which simply express that since there is no dissipation, the populations of optically coupled levels and the modulus of the Bloch vector are constant. Considering excitation by a monochromatic field a/at + 0 and transforming to the electron-hole representation n,,(k) = n,(k) = n, and n,,(k) = 1 n,(k) = I - nk, we get (6, - hw)p, - (1 - 2nk)Ak= 0 and .Fm(ptAk)= 0, where ck = c,(k) - &,(k). The last equation expresses that there is no absorption. It is worth noting that in the small excitation regime, the conservation of the Bloch vector modulus implies that 8, z Ipk12.The eigenvalues of fi, give the dispersion of the “dressed bands:

As mentioned before, the curvature of these “dressed bands is different from

that of the unexcited semiconductor. This can be viewed as an “electronic polaron” effect, whereby the e and h drag with them the cloud of short-lived [ T % h / ( C k - ho)]virtual e-h pairs, hence becoming heavier (Perakis and Chemla, 1994). The Hamiltonian H = & i i k can be diagonalized by a canonical transformation 0,. In the new basis, the density matrix 0,EikQ:has only one nonzero matrix element equal to 1 in the ground “condensed” state. The

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solutions are

and the “gap” equation is

The approach sketched in the preceding paragraph is a mean field theory of the laser-excited semiconductor, where P k and nk are the order parameters. They play, respectively, the same role as the pair amplitude and the mean occupation number in the BCS theory of superconductors. Indeed, Eqs. (33) and (34) are formally identical to the BCS equations describing superconductivity (Schrieffer, 1988) and to those describing the Bose condensation of excitons (Keldysh and Kozlov, 1968; Comte and Nozieres, 1982) with two modifications: (1) the chemical potential is replaced by the photon energy, and (2) the gap equation contains an extra term, the Rabi frequency. The first change expresses that each virtual exciton is created by a photon and the second that the condensation is not spontaneous but induced by the EM field, In the same spirit it is worth noting that Eq. (27) is the analogue of Anderson’s “pseudo-spin Hamiltonian” in superconductivity theory. In order to describe pump and probe experiments, one has to consider that the applied field is comprised of two parts, the strong pump field E(t) and the weak probe field 6E(t). The former gives rise to a renormalized semiconductor “ground state,” described by Eqs. (33) and (34), whereas the linear response to 6 E ( t ) yields the corresponding renormalized “excitation spectrum,” which is blue shifted and thus exhibits the OSE. The theory also predicts a small nonlinear and coherent gain below the pump central frequency that involves processes of high order in the applied field (SchmittRink and Chemla, 1986; Schmitt-Rink et al., 1988). All these results stress the profound effects that the Coulomb interaction has on the optical properties of semiconductors. The physics of these materials excited by a laser field is more closely related to that of condensed manybody systems than to that of laser-excited atoms. Another description of the excitonic OSE has been proposed ‘(Combescot and Combescot, 1988; Combescot, 1992). Although starting from a different point of view, it yields results that are equivalent to those of Schmitt-Rink and Chemla (1986, 1988) when the biexciton contribution does not play an important role.

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Linearizing Eqs. (31) and (30) with respect to SE(t), and considering only the effects of the leading 1 s resonance to get analytical results (Schmitt-Rink and Chemla, 1986; Schmitt-Rink et al., 1988), one finds that the exciton peak is shifted by

where NFF is the saturation density due to excitonic PSF mentioned in Section 11. The first fraction in Eq. (35) reproduces the atomic results (Eq. 21); the second gives the changes due to the excitonic structure. The dependence of the OSE shift on dipole matrix element was confirmed by comparing the shifts experienced by hh-X and lh-X in the same measurement (von Lehmen et d.,1986). Its dependence on the exciton relative motion wavefunction was studied by distorting &,s(r = 0) using an electrostatic field applied perpendicular to the plane of the QW. A strong reduction = 0)l2 decreases, in of the excitonic OSE has been observed as agreement with theory (Knox et al., 1989; Chemla ef al., 1989). A more complete treatment, using the Coulomb Green's functions for solving the formalism of Schmitt-Rink and Chemla (1986) reveals that the condensation of virtual excitons is distributed over all the bound and unbound states (Zimmerman, 1988a, 1988b, 1988~).The relative occupation is a sensitive function of the detuning and pulse duration. Interestingly, it is related to the real part of the susceptibility in contrast with the cases where real excitons are generated and where the occupation is related to the imaginary part of the susceptibility (Zimmerman and Hartmann, 1988). Numerical calculations based on the theory of Schmitt-Rink and Chemla (1986, 1988) predicted that in 2D and for pump detunings of about IOR,, low-intensity nonresonant excitation should produce a pure Stark shift of the 1s exciton without any loss of oscillator strength, while in 3D the oscillator strength increases slightly with pump intensity (Schafer et al., 1988; Schafer, 1988; Ell et al., 1988; 1989). The physical origin of this behavior is that the Stark shift of the band gap is always larger than that of the IS exciton because of the larger spatial extent of scattering states. This corresponds to an effective increase in the 1s exciton binding energy and thus in its oscillator strength, which can overcome the decrease due to phase space filling by the virtual e-h pairs. An example of DTS at At = 0 seen in a Q W sample excited 50meV below the 1s hh-X for a low pump excitation, 30 M W/cm2, is shown as a solid line in Fig. 8. In the case of a pure shift, the DTS should have exactly the same line shape as the derivative of the linear absorption da(w)/dw, which is shown as a dashed line in the figure.

-

205

3 OPTICAL PROCESS~SIN SEMICONDUCTORS 0.1

h

8

Y

E 2

8 P

o

c

.-0

1a Q

z

c

E Q E

n

- 0.1 I

I 1.565

l

i

l

1.575

l 1.585

I

l

l

l

Photon Energy (ev)

FIG. 8. At low excitation and large detuning, the excitonic optical Stark effect corresponds to a pure shift without change of oscillator strength. This is shown by the comparison of the differential transmission spectrum,measured in a QW sample excited 50meV below the 1s hh-X for a pump intensity -30MW/cm2 (solid line), with the u-derivative of the linear absorption spectrum (dashed line) (Knox et al., 1989).

Indeed, the two line shapes are identical. In addition, the integral of the DTS around to the resonance is zero, as it should be for pure shift. As the pump intensity is increased, the DTS profile becomes indicative of both a shift and broadening. This is due to the finite bandwidth of ultrashort pulses whose Fourier components have different detuning and amplitude, leading to an “inhomogeneous broadening” of the absorption spectrum, which increases with intensity (Schafer et al., 1988; Schiifer, 1988). An alternative interpretation of this broadening is that for ultrashort-pulse excitation at small detuning, the effective masses of the “dressed” bands experience a significant and time-dependent variation. For the “slow” degrees of freedom, such as an e orbiting an h in a bound state, the time-dependent masses induce a strong dephasing. Very recently, the coherent nonlinear gain below the pump frequency has been observed (Likforman er al., 1997).

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The OSE also was observed in modulation-doped samples, where excitons are not present, and the absorption edge exhibits an FES. In ultrashortpulse experiments carried out under the same conditions as for undoped samples, together with the blue shift there was observed a small but significant optical gain just below the FES and a much smaller reduction in the optical absorption strength (Brener et al., 1995). The FES OSE is much more complicated to treat because the reaction of the whole Fermi sea must be accounted for. The low-lying excitations of the Fermi sea are slow; they cannot adjust adiabatically to the rapid effective mass changes induced by the virtual interband transitions. The combination of the reduced “electronic polaron” enhancement of the effective masses and the time dependence of gap E , explains, qualitatively, the small gain and the changes in the FES absorption (Perakis and Chemla, 1994; Perakis, 1996; Perakis et al., 1996). I t is worth noting how much the formalism and the interpretation of the experiments discussed earlier are different from that of atomic systems. Having introduced the main manybody concepts, we can now proceed to the case where real carriers are created.

V.

Fundamentals of Two-Particle Correlation Effects Involving Real Electron-Hole Pairs

This section discusses the fundamental mechanisms that are responsible for nonlinear optical processes in semiconductors when real e-h pairs are created. The first experiments performed with ultrashort-pulse excitation at or slightly above exciton resonances were concerned with the effects that a gas of excitons or an e-h plasma have on the excitonic absorption in bulk (Fehrenbach et al., 1982; 1985) and in Q W structures (Chemla et al., 1984; Peyghambarian et al., 1984; Knox er al., 1986; Hulin et al., 1986; Chemla et al., 1988). The loss of exciton oscillatior strength in the presence of an e-h plasma seen in the bulk was first attributed to screening. It was the interpretation of pump/probe experiments performed on quasi-2D Q W structures, where the effects of screening are reduced, that helped identify Pauli blocking of exciton transitions as important (Schmitt-Rink et af., 1985; 1989). Interestingly, it was found that the reduction of oscillator strength occurred as the absolute energy of the exciton remained constant in 3D (Schultheis et a/., 1986; Fehrenbach et a!., 1982; 1985), whereas in 2D it was accompanied by a nonnegligible blue shift (Peyghambarian et al., 1984; H u h et al., 1986). Furthermore, the relative magnitude of these effects depended strongly on the nature of the excited species, exciton gas, or e-h plasma and on their temperature ?; resulting in complicated dynamics. The

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e-h pairs appeared to be more effective than an exciton gas in reducing the band-edge absorption at low temperature but less effective at high temperature. Although a comprehensive theory was not available at that time, the constant value of the exciton energy in 3D was explained as near-perfect cancellation of Pauli blocking (hard-core repulsion) and screening (Fehrenbach et al., 1982; Haug and Schmitt-Rink, 1984). In 2D, since screening is strongly reduced, this compensation does not occur, and the Pauli blue shift dominates (Schmitt-Rink et al., 1985; 1989). Analytical results for the change in the exciton oscillator strength& in the leading order in the density N were obtained in reasonably good agreement with experiments:

+

where NFF accounts for the PSF of Here, N;' = (NFF)-' (Nf.XCH)-', the states out of which excitons are made and NFXCHaccounts for the change in the exciton relative motion wavefunction due to screening. It follows from Eq. (10) that (NFsF)-' = Xk [ n e ( k ) + n h ( k ) 1 4 a ( k ) / 4 a ( r = For an exciton gas, n,(k) = n,(k) a l4,(k)I2 and (NFF)-' is proportional to the exciton volume (411a2/3 for n = 1s in bulk and 2na& in QW). For the e-h plasma, NPFdepends on the temperature through the overlap between the plasma distribution n,(k) and n,(k) and &,(k) (Schmitt-Rink et al., 1985; 1989). At low temperature, the carriers occupy the band minima and efficiently block the transitions at k < a;'; at high temperature, they spread out, the overlap decreases, and N< a R,/kB '7: For the same reasons, NFXCH follows similar trends. The ultrafast evolution between these different regimes is shown in Fig. 9. Here the DTS spectra of a 10-nm GaAs QW structure excited about 20meV above the exciton resonances are presented as the pump/probe time delay At is varied from -100 to 200fs by 50-fs steps (Knox et al., 1986). At early time delay one sees a spectral hole around 1.513eV that follows approximately the pump spectrum and a more complex DTS signal at the exciton resonances around 1.49eV indicative of broadening and loss of oscillator strength. The spectral hole is due to the PSF induced in the continuum states by the nonthermal distributions of photocarriers generated by the pump. As shown in the inset of Fig. 9 for another experiment, the spectral hole is always shifted as compared with the pump spectrum. The line shape of the spectral hole and its location relative to that of the pump are discussed below. The signal at the exciton peaks is due to a combination of collisional broadening (see Section VII) and coherent excitation of the

'

D.S. CHEMLA

208

1 0.04 *

0.02

c

0.00

.

-

0

(0

8 S

0

e 0.02 9

9-

6

E

C

2

m s: c3

0.00

Energy (eV)

FIG.9. Differential transmission spectra versus time delay measured on a GaAs quantum well structure at room temperature excited about 20meV above the exciton resonances (Knox et ul., 1986). One sees clearly the Pauli blocking of the absorption by the nonthermal carrier population as it is created and thermalizes. The signal at the exciton resonances increases as the carries fill up the bottom of the bands. Inser: Comparison of the pump spectrum with a differential transmission spectrum at Ar = 0 showing that the spectral hole is downshifted as compared with the pump spectrum.

resonances. As Ar increases, the spectral hole in the continuum moves down in energy and smooths out until At x 200fs, where it has acquired an exponential profile reminescent of a Maxwell-Boltzmann distribution. This is interpreted as due to the cooling down and thermalization of the photoplasma. Simultaneously, the signal at the excitons increases because PSF and screening increases with the overlap of the plasma distribution with the exciton wavefunctions 4,(k). For small time delays, the spectral hole is always seen shifted with respect to the laser spectrum, as detailed in Fig. 10 (Foing et a/., 1992). This very interesting coherent effect was investigated experimentally (Foing et al., 1992; Mycek el a/., 1996) and theoretically (Zimmermann, 1988a; Tanguy and Combescot, 1992). The mechanisms responsible for the FES, discussed in Section I, generalized to a nonequilibrium population (Foing et al., 1992; Mycek et a/., 1996; Tanguy and Combescot, 1992) give a qualitative understanding of the observations. The quasi-instantaneous e-h distribu-

3 OPTICAL PROCESSES I N SEMICONDUCTORS

209

I

1

:\

:' 15

158

U

8 - -0.1 Q

J

?

'

I

1.66

' .

I

/

-0.1 J J 1.54

158

1.62

1.66

1.7

EnagY(eV1 FIG. 10. Pump/probe differential transmission spectrum measured for a set of time delays in a GaAs sample at low temperature (solid lines) as compared with the pump spectrum (dotted lines) (Foing et al., 1992). The line shape and the shift are due to excitonic effects at the two edges of the transient populations generated by the pump. The inset shows the laser spectrum and the sample absorption spectrum.

tions created by ultrashort laser pulses have two edges that are not sharp, since they follow the laser pulse spectrum. Immediately on their creation, however, the carriers are coherent and thus can participate in FES-like processes. Manybody theories have been developed to describe these mechanisms (Tanguy and Combescot, 1992); for an intuitive discussion, let us neglect the valence-band dispersion and assume that e-e interactions do not perturb significantly the h-Fermi sea interaction. We can then consider that the ultrashort laser pulses create a nonthermal electron distribution ne(&- E ~ t), centered at energy E~ that one can describe as the sum of two parts: ne(&- c0, t ) = nce(&,t ) - neh(&,t), a negatively charged distribution of conduction-band electrons ce, nce(&,t) = O(E - E,,) + O ( E-~E ) x ne(&- E,,, t), and a positively charged distribution of conduction band holes ch, nch(&, t) = O(E- c0) x [l - ne(&- go, t)]. These two distributions have a steplike Fermi profile, and one can apply to them the static FES theory (Mahan, 1967; Nozieres and de Dominicis, 1969) (1) as long as the nonthermal distribution remains coherent and (2) with opposite sign for the cb-e and the

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cb-h distributions, i.e., at the high energy c2 and low energy edges. As for the case of the static FES, the singularities are smeared out because of processes neglected in this oversimplified discussion and appear as resonances. Altogether, the dominant physics is that the absorption is enhanced close to c2 because of the attraction between the cb-e-Fermi sea and the photogenerated valence hole, whereas it is reduced close to e l because of the repulsion between the cb-h-Fermi sea and the valence hole. Let us note that, in the same spirit, excitonic effects at the edges of nonthermal Gaussian distributions were obtained in the theory of pump/probe DTS involving a transient e-h population (Zimmerman, 1988a). Similar considerations show that the same resonant features occur in the emission process. The dynamic FES at both edges of the nonthermal distribution lead to absorption enhancement and emission reduction, i.e., to a blue shift of the spectral line shapes of the absorption and emission spectra, as compared with that of the pump laser pulses. A crucial point on which all these arguments is based is that of the coherence of the nonthermal distribution; the energy shift lasts only as long as the coherence is maintained. This issue was investigated by a comparative spectrotemporal analysis of the time sequence for generation of nonthermal distributions in pump/probe and self-diffraction four-wave mixing (FWM) experiments (Mycek ei al., 1996). The self-diffraction FWM technique is aimed at measuring the simplest nontrivial coherent emission. Two laser pulses, E , ( t ) and E2(t), separated by a time delay, At = t 2 - t,, and propagating in the directions and interfere via some nonlinearity in a sample. They generate several nonlinear polarization waves that contain one contribution, P,(t, At), emitting photons in the background-free direction = 2x2 The excitation configuration is exactly the same as for pump/probe measurements, although, in the latter case, it is the change in the transmission of thex;, pulses that is measured. It is important to emphasize the different time sequences for the generation of the first-order polarization p"', second-order population d2), and third-order polarization in the two experiments. For FWM, p"j(xl) a n")(X2 -2,) a F m [ p " ) ( x , ) * E ( ~ , ) ] and , pt3)(z,)a E(x,)n'2)(x2-Il),whereas for pump/probe, p")(x2) a E(x,), n ( 2 ) ( l= 0) 3: 3 m [ p ( 1 ' ( x 2 ) * E ( ~ , )and ] , pf3)(x1) a E(x;,)n'')(x= 0). Our convention for positive delay, Ar = t, - t , > 0, refers to the usual one for the polarization decay of a two-level atom. The dynamic FES is expected to occur on a time scale where incoherent screening starts to become effective. In the experiments of Mycek et al. (1995-1996), the FWM and pump/probe power spectra measured versus Ar show a shift of the instantaneous frequency immediately after the nonthermal second-order distributions d2)(x2 - 7,) and n(')(K = O), respectively, are created, i.e., for the opposite time sequences

x2

xs

EG1),

-xl.

x,

3 OPTICAL PROCESSES IN SEMICONDUCTORS

211

z2

between the pulses in direction and TI.This demonstrates that the blue shift associated with the dynamic FES does not depend on the particular time ordering of the pulse sequence; it is only related to the coherent part of the manybody interaction, i.e., as long as the n(’)’s are coherent. Importantly, the disappearance of the blue shift is still seen well after the conoentional relaxation time T, (Becker et al., 1988; Bigot et al., 1991), clearly indicating that some correlation in the e-h system survives for rather long times. In fact, the experiments show that the loss of coherence in a dense medium is far too complex to be described by a single parameter such as T, (Mycek et al., 1996), as is discussed in Section VIII. of the The first investigations directly testing the relaxation time excitonic polarization were performed by degenerate FWM. In the case of a homogeneously broadened two-level atom, the FWM signal is emitted immediaten after the second pulse and corresponds to “free polarization decay” (Yajima and Taira, 1979). For inhomogeneously broadened atomic lines, the FWM signal is delayed by At after the second pulse and corresponds to a “photon echo” (Allen and Eberly, 1987). FWM techniques have been applied extensively to atomic and molecular systems. In the two-level atom case of free polarization decay, P,(t, At) is zero for At < 0 and exhibits a simple exponential decay for At > 0. For this reason, the easiest and most commonly used measurement technique for atomic-like systems is to time integrate the FWM signal with a slow detector as At is varied to determine the so-called time-integrated FWM (TI FWM):

For a two-level atom, &(At) reproduces as a function of At the same temporal behavior as IPs(t, At)12 versus t at any fixed At. Because of the historical background of atomic and molecular physics and the simplicity of the two-level atom results (Yajima and Taira, 1979), the early ultrashortpulse investigations of FWM in semiconductors concentrated on the dephasing of resonances and were analyzed using that model (Schultheis er al., 1985; 1986a; 1986b). It was deduced from the decay of ST1versus At that the exciton dephasing time was in the picosecond time scale. These experiments were extended to study the effects of temperature and the density of exciton gases and e-h plasmas on the exciton dephasing time (Honold er a!., 1989a; 1989b). Again, early analyses were performed according to atomic models as sketched in Section 111, and we will revisit this issue in Section VII. A qualitative difference with the ideas commonly accepted in coherent spectroscopy was observed when very high quality heterostructures, with

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-

homogeneously broadened exciton resonances, were probed wth 100-fs pulses (Leo er ai., 1990). As shown in Fig. 11, TI FWM experiments revealed a very strong STI(At)signal for At c 0, extending at least as far as 20 times the laser pulse duration before A t = 0. As seen in the figure, temperaturedependence study demonstrated that the rise time of S , = ST,(At c 0) is exactly half the decay time of the “regular” signal S& = STI(At> 0). This direct contradiction with the atomic theories, which always predict that S+, 3 0 identically (Yajima and Taira, 1979), forced a reevaluation of the analysis of coherent processes in semiconductors. As shown below, the Coulomb interaction induces nonlinearities that have a behavior qualitatively different from that of two-level atoms and are responsible for the nonzero value of SG. The two-parabolic-band model formalism described in Section 111, with the addition, in Eq. (26), of phenomenological relaxation and dephasing times describing the interaction with phonons, gives a good starting point

3

1 h

4

-2

0

2

4

6

-Dehv@5) FIG. 1 1 . Time-integrated self-diffracted four-wave mixing signal from a homogeneously broadened exciton resonance in a 17-nm GaAs quantum well structure (Leo et al., 1990). The negative time delay signal is due entirely to the Coulomb interaction.The rise time of the signal for Ar < O is exactly half the decay time for Ar >O.

3 OPTICAL PROCESSESIN SEMICONDUCTORS

213

for discussion (Wegener et al., 1990). It is instructive to write the coupled equations satisfied by the density matrix elements. For the off-diagonal and diagonal terms, Eqs. (30) and (31) in the e-h representations, n,(k) = n,(k), n h ( k ) = 1 - n,(k), we have respectively,

and

These equations deserve several comments. First, we should note that in Eq. (38) we have identified the “observed” band gap Eg with the energy difference &,(k = 0) - &,(k = 0) C,. &,k*. Thus, within the very restrictive HF/RPA discussed in Section 111, this indicates that the gap energy includes the Coulomb interaction of the full valence band electrons n h ( k ) = 0 and ne(k) = 0. The sources of nonlinearity in Eqs. (38) and (39) are due to the excited photocarriers n h ( k ) # 0 and ne(k) # 0, but their origin stems from the same potential &,k, that determines the gap. Hence my remark in Section 11: The Coulomb interaction causes zero-order effects; this is, in fact, a general result substantiated by much more thorough theoretical treatments (Hybertsen and Louie, 1985; 1986; Louie, 1997). If the right-hand side of Eq. (38) is put equal to zero and the steady state is assumed (dp,/at + 0), one recovers the k-space exciton Wannier equation (Eq. 5), so Eq. (38) includes all the “excitonic” effects. If is put equal to zero in Eqs. (38) and (39), one recovers the optical Bloch equations for the independent two-level atom model of the “atomic” picture. The nonlinear source term on the right-hand side of Eq. (38) is comprised of two parts. The first one expresses the reduction of the Rabi frequency because of Pauli blocking, and it is active for all material systems made of Fermions, atoms, molecules, or solids. It appears as a coupling between electric field E(t) and the populations n,(k) and n,(k). The second term expresses the Coulomb coupling between polarization P k and populations n,(k’) and n h ( k ’ ) . This term is new; it appears only in condensed matter. Because of the consistent treatment of the self-energy and vertex corrections,

+

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D. S. CHEMLA

it vanishes for k = k , avoiding unphysical divergences and translating the fact that a plane wave does not interact with itself. Finally, Eq. (39) expresses that the populations are generated by the total field Ak of Eq. (29). In this chapter I shall call the HF/RPA Coulomb nonlinearities the bare Coulomb interaction (BCI). The set of coupled equations (Eqs. 38 and 39) is called the semiconductor Bloch equations (SBEs). Over the last decade, the SBEs have been applied very successfully to explain a number of nonlinear optical processes in semiconductors (Schafer, 1988; 1993; Lindberg and Koch, 1988-1; Haug and Koch, 1993). In the k-space representation, numerical solutions of the SBEs, with a complete description of the energy-band structure, have been applied successfully to realistic materials and heterostructers (Schafer, 1993). They also have been investigated in the r-space representation (Stahl and Balslev, 1987; Balslev et al., 1989), where interesting spatiotemporal aspects of the polarization dynamics are better expressed (Glutsch et al., 1995-1). Coming back to the experimental results of Leo et a/. (1990), let us see how they can be explained by the SBEs. The processes responsible for the FWM signal in the direction k, are at least third order in the field. They can originate from one of the two nonlinear sources of Eq. (38). For ultrashort pulses, the polarization and population components rise with the fields and then decay exponentially. Thus the PSF source cc [n,(k) + n,(k)]pkE(t) generates the FWM signal only for the sequence where the field E(x,) overlaps with the second-order population n(”(x, -Il), i.e., for At > 0. On the contrary, the Coulomb source cc h.k.pk[n,(k) + n,(k)] is nonzero for both At < 0. Furthermore, since in the f 3 ) regime n(’)(X, -!,I K . Y m [ p ( ’ ) ( x , ) * E ( ~ , )the ] , rise time of S i is twice as fast as the decay time of S;,. Therefore, in the experiments of Leo et af. (1990), the observation of S , and the value of its rise time are a direct manifestation of Coulombmediated manybody effects. The SBEs suggest further interesting aspects of FWM. Since the PSF due to Pauli blocking is instantaneous but the polarization needs to build up to make a significant contribution, the “time resolved” FWM signal (TR FWM), measuring the “absolute time” dependence of the polarization P,(t, At), at fixed At, STR(4 At) a IP,(t,

Wl’

(40)

is expected to be highly nonexponential and is comprised of two contributions, one instantaneous, due to PSF, and one delayed, due to Coulomb manybody effects. This was indeed observed, leading to novel features of coherent wave mixing not seen in atomic systems (Mycek ef af., 1992; Weiss et al., 1992; Kim et al., 1992). These effects can be significant, as shown in

3 OPTICAL PROCESSES IN SEMICONDUCTORS I

215

GaAsMQW 10K (a)

'

co-polanzed

C-

-

- 1 0 1 2 3 4 5 6

Time t (ps)

FIG. 12. Time-resolved self-diffracted four-wave mixing signal from a 17-nm GaAs sample showing that the emission due to the Coulomb interaction can be so much delayed that it appears as a separate pulse (Kim ef al., 1992).

Fig. 12. In high-quality materials at low temperature, the T R FWM signal is very substantially delayed and even appears as a pulse well separated from exciting laser pulses. At high temperature, as the exciton density is increased, excitons are ionized, generating free carriers that screen the Coulomb interaction and hence modify the relative contribution of the two sources of nonlinearities. A careful study and analysis of TR FWM experiments in GaAs QWs at room temperature gave an accurate measure of the relative strength of PSF and BCI as a function of the photocarrier density (Weiss et al., 1992). In Fig. 13, the ratio of the BCI/PSF contributions is plotted versus

216

D. S. CHEMLA CI)

c

'3

C

I

a

2.5

0.5 -

I

I

I

.

,

, . . . 1

fic. 13. Ratio of the contribution of the bare Coulomb interaction and the Pauli blocking (phase space filling) nonlinearities as a function of the density of carrier excitations (Chemla and Bigot, 1995).

N,,,m&,, the number of photogenerated carriers per Q W exciton area. The plot exhibits a remarkably sharp transition from a BCI-dominated regime for N,, x miw< 1 to a PSF-dominated one for N,h x naiw > 1, quite evocative of a phase transition. It is interesting to express the SBEs in the exciton basis. Since the wavefunctions of the bound and unbound states +,(r) form a complete basis, any function of r and t, f(r, t), can be written as

f(r, t ) =

2fz(tW.(r) 1

with L(l)=

I

W ( r , tMa(r)

(41)

and the polarization is expressed as

Applying Eq. (41) to the diagonal and off-diagonal elements of the density matrix yields

3 OPTICAL PROCESSES IN SEMICONDUCTORS

217

(43)

and (44)

where

KBy=

s

dr’drV(r‘)q5z(r)4S(r‘)cjy(r - r‘)

is the nonlocal Coulomb coupling between excitons. If we restrict ourselves to the linear regime, Eqs (43) and (44) are similar to that of a two-level atom with the substitution p + p$.*(r = 0). Since the polarization is obtained by multiplying the polarization amplitudes by 4 J r = 0), in the early days of the study of excitons it was commonly accepted that the only effect of BCI was the excitonic enhancement of the oscillator strength -+ 1pI214,(r = 0 )l’ consistent with Elliott’s formula (Eq. 7). Obviously, in the nonlinear regime, excitons are sensitive to both PSF and BCI. It is worth noting some aspects of the nonlocal BCI between excitons: (1) it is active even when only one e-h pair is excited but is distributed over several exciton states, as in the case of ultrashort-pulse excitation, and (2) it vanishes exactly for a = fi = y, showing that a single exciton does not interact with itself. It is also interesting to note that the first nonlinear polarization P 3 )a x(3) contains a PSF term cc lpI414,(r = 0)l’ and BCI terms a lp1414a(r = O)I3 and a 1p1414ax (r = O)I4, showing that the exciton internal structure does not affect the two sources of HF/RPA nonlinearity in the same way. A useful and intuitive model can be deduced from Eqs. (43) and (44). Assume that in the case of ultrashort-pulse excitation with a significant linewidth one can replace the polarization (Eq. 42) by an “average” 9 so that the sums in Eqs. (43) and (44) are written as averaged as well. Assuming, furthermore, that excitation is low enough that one can take ni(k) x Ipk12,9’is found to satisfy the nonlinear Schrodinger equation:

218

D. S . CHEMLA

where PSis a saturation parameter, and *Y is an effective Coulomb coupling (Wegener et a/.,1990; Schmitt-Rink et al., 1991). i call this approximation the efective polarization model (E PM). 9’ behaves like a harmonic oscillator driven by two source terms. These express the dual character of laser-excited semiconductors.The first source term translates the “atomic” character of the optical transitions and has its origin in the Pauli blocking saturable electron-photon coupling. The second is a Coulomb-mediated self-interaction, which has the same form as that of the order parameter in the Ginzburg-Landau theory of superconductivity and has its origin in the electron-electron manybody coupling. Equation (45) captures the essential physics of the light-semiconductor interaction at the HF/RPA level, and it was found very useful for intuitively explaining a number of experiments (Wegener et al., 1990; Schmitt-Rink et al., 1991; Weiss et al., 1992). In the following sections I will use the EPM and its generalization to discuss the physics behind a number of interesting observations. This is very convenient for giving an intuitive picture. However, for accurate simulation of experiments, it is necessary to use the full numerical solutions of the SBEs, including the actual band structure and spin-selection rules of each particular sample.

VI. Applications Spectroscopy and Dynamics of Electronic States in Heterostructures

Modern optoelectronics makes extensive use of semiconductor heterostructures, and very often electronic and photonic devices operate in conditions of high density and high field. A generic question of this field of research is to understand and control the electronic states and their dynamics in these artificial structures. In this respect, time-resolved nonlinear optical spectroscopy has proven to be a very powerful tool, much more versatile than such conventional techniques as photoluminescence. The interpretation of many important experiments with ultrafast dynamics requires a correct description of the interplay between PSF and BCI. This section reviews some of them. A first example is the quantum beats (QB) observed in TI FWM when the Ih-X and hh-X of a QW structure are simultaneously excited by ultrashort pulses (Feuerbacher et al., 1990 Leo et al., 1990 1991; Koch et ul., 1996). As shown in Fig. 14, when the central frequency of the excitation is tuned between the two excitons, both S&(At) and S , ( A t ) exhibit a strong modulation in time. The period is related to the lh-X/hh-X energy splitting *&, - Ehh = hfilh-hh by ToB 2 h / f i l h - k h . If the beat period is in agreement with an atomic-like three-level-system (3LS) model (Leo et al., 1991), the

3 OFTICAL PROCESSES IN SEMICONDUCTORS

219

Time Delay (ps) FIG. 14. Quantum beats between the heavy-hole and light-hole excitons seen in the time-integrated four-wave mixing signal measured on a 15-nm GaAs quantum well structure when the two excitons are excited simultaneously(Leo er al., 1991).

large S,(At) indicates that BCI is active as well. The FWM is also found to be very sensitive to the polarization of the laser pulses. The combination of a large &(At) and polarization selectivity indicates that one must include both the Coulomb interaction and an adequate band structure for explaining the data. The polarization selection alone can be accounted for by phenomenological atomic-like models with six levels reproducing the near-bandedge spin symmetry of Fig. 1 (Schmitt-Rink et al., 1992). The correct interpretation is given by the SBE formalism with the full six spin-degenerate bands Luttinger Hamiltonian (Schafer, 1993). I will analyze more in detail the important question of FWM polarization selection rules in Section VII. For the moment, we must be satisfied with a qualitative discussion. Using the band structure sketched in Fig. 1, we see that it is possible to build up four exciton manifolds for the transitions hh -+ e and Ih -+e excited by photons with polarization o*. Introducing the mixed band-spin indices v = (hh, +), (Ih, +), a straightforward generalization of the EPM is

220

D. S. CHEMLA

where the PSF due to excitons sharing a common band is characterized by the saturation parameter B,,,,., and the BCI coupling between two exciton species is characterized by the parameter *Yvv..It turns out that at the HF/RPA level, the BCI is diagonal with respect to the different bands due to the orthogonality of the different spin states in the conduction and valence bands. Thus, for FWM in a linear parallel polarization configuration (with all photons 11-polarized), the four exciton species are excited. They are quantum mechanically coupled by PSF, and BCI coupling is active within each spin manifold. Thus the oscillations in the FWM signal are true quantum beats. Conversely, in the cocircular a*/a*-polarization configurations, the spin-polarized h X * and hh-Xi do not share a conduction band, and they are not BCI-coupled. Thus, in these polarization configurations, any beat seen in &,(At) originates from a polarization interference (PI), unless processes beyond the SBEs are active. As we will see in Section VII, the coupling between lh-X* and hh-X* is in fact a signature of four-particle interaction processes not accounted for at the HF/RPA level. Finally, in the cross-linear case (I-polarization configuration), I will show in Section VII that among the processes beyond the SBEs, only the exciton-exciton exchange is active, and thus the FWM signal is weaker, whereas for the o-/a+-polarization, one does not expect to see an FWM signal at any order (Bartels and Stahl, private communication). These investigations triggered several experiments aimed at distinguishing the QBs within a single multilevel quantum mechanical system from the interference in the emission of independent two-level atoms (Koch et al., 1992; Lyssenko et al., 1993). This latter can be considered as the simplest case of inhomogeneous broadening, and in TR FWM experiments, the rephasing of the different emission frequencieshas the same time dependence as for a photon echo. Therefore, the maxima of the spectrally resolved FWM signal,

vary as max[ST,(t, At)] = 2At + 47tn/&-fi. The QBs, on the contrary, follow the same time dependence as free polarization decay; i.e., max[STR(t, Ar)] z At + 2 7 t n / f i l h - h h . This was verified in an elegant experiment (Koch et ui., 1992) by comparing the TR FWM from two QW samples. One consisted of only one type of QW and thus exhibited hh-X/lh-X QBs, whereas the other sample had two types of QWs electronically separated, with distinct exciton transitions, and gave only PI. A comparison of the ST1(Al)in the two cases is shown in Fig. 15 (Koch et al., 1992), where the two slopes are easily distinguished. The different time behavior of QBs and PI has its counterpart in the frequency domain (Lyssenko et al., 1993). The component

(a) Quantum beats

(b) Polarization beats w .y.

PY

E4H E

il

3

-

Real time [ps]

.-ii

=3

Real time [ps]

-

g 1.6 .-!i Y

2

ai

....

. 0

1.0 2.0 lime delay [ps]

0.0

IrJ: 0.8

2

0.0

0.0 0.4 0.8 Time delay [ps]

FIG. IS. Distinction between quantum beats and polahtion interference (Koch et a!., 1992). Top graphs: The amplitude of the time-resolved four-wave mixing signal is plotted as a function of the absolute time f for a series of time delays At in the case. of (a) quantum beats and (b) polarization interference. Lower graphs: Position of the signal maximum in the t = At plane is the solid line for the two cases; the dashed lines show the t = 2At and the t = At slopes.

222

D. S. CHEMLA

of S,(w, Ar) at a given frequency usshows oscillation as At is varied. By scanning ws across the resonances, it is found that the oscillation pattern remains unchanged for QBs, but for PI it experiences a n shift, and the signal amplitude vanishes exactly at the center of the two resonances. An important aspect of QBs is related to the uncertainty principle. As for any quantum mechanical effect in a single system, the frequency shift during a QB cannot be instantaneous and must satisfy AE x At > h. This was actually measured in experiments where both the amplitude and the phase of the FWM signal were determined (Bigot et al., 1993; Chemla et al., 1994-11; Chemla and Bigot, 1995). Figure 16 shows the “time-energy” picture of QBs obtained under conditions where the spectral weights of the hh-X and lh-X contributions were equalized [Fig. 16(a)]. The QBs appear beautifully in the interferometric first-order autocorrelation (AC FWM):

Figure 16(b), with a 230-fs beat period, corresponds to the hh-X/lh-X splitting seen in the spectrally resolved FWM. The phase of the FWM signal relative to the laser [Fig. 16(c)] shows that the emission starts approximately in coincidence with the laser, which in this case coincides with the lh-X. Then, around 120fs, it experiences an abrupt n shift when it moves suddenly to the hh-X, where it remains until the next beat. The K shift does not occur instantaneously but takes about 50fs to be completed. The Q B “duration” is more precisely determined in Fig. 16(d), where a set of fringes at the center of the AC FWM is compared with a set of fringes close to the first node. One can actually count the number of fringes it takes to complete the 7c shift. The frequency modulation is very fast, AE x At = (1.4 & O.l)h, yet still above the fundamental quantum limit. QBs also were observed and investigated in other systems, including magnetoexcitons (Bar-Ad and BarJoseph, 1991; Wang er al., 1992), excitons localized by interface roughness in Q W structures (Gobel er al., 1990), and unbound e-h continua states (Joschko er a/., 1997). Ultrafast nonlinear optical spectroscopy also has found useful applications in the study of electronic transport in coupled-layer heterostructures, such as resonant tunneling of electronic wave packets in double-QW systems and Bloch oscillations in superlattices. A double-QW system that consists of two QWs with thickness L, > Lz separated by a very thin large-gap barrier layer can sustain tunneling of e wave packets between the two QWs when the conduction subband energy levels in the two QWs are brought in coincidence, say, by application of an

3 OPTICAL PROCESSES INSEMICONDUCTORS

223

FIG. 16. Time-energy picture of the four-wave mixing signal showing that the shift in emission frequency in a quantum beat is not instantaneous and satisfies the uncertainty principle (Chemla er al., 1994-11). (a) Spectra of the laser (dotted line) and four-wave mixing signal (solid line). (b) Interferomeric autocorrelation (AC) of the four-wave mixing signal. (c) AC fringe spacing relative to the reference laser showing the sudden n shift during the quantum beat. (d) Detail of the interferomeric AC at the center of the profile and near the first node.

electrostatic field. The electrostatic levels of the combined QW system are approximately the symmetric and antisymmetric combinations of the isolated QW levels and are separated by h6RS,,,. For properly chosen L, > L:, the hole levels are not coupled and remain localized in each QW. An electronic wave packet prepared in one QW will oscillate between the two QWs with a period 7e5,,,= 2n/6i2s2,,s.The electron motion can be detected by probing the interband transition between the hole localized in one Q W and the combined electronic levels, because when the electron is in the same QW as the hole, that transition is blocked by PSF. This program was actually performed in pump/probe and FWM experiments (Leo et al., 1991). Figure 17 presents the DTS spectra measured on a heterostructure that consisted of double-QWs within a pin diode; the resonance condition was achieved by applying a reverse bias to the pin diode. The DTS shows periodic oscillation with period rose z 1.3ps close to the nominal value corresponding to

224

D. S.CHEMLA

1

W Excitation

140/25/100A

I

10 K

-I

F [kV/cm] -1

0

1

I 2

3

Time delay Ips] FIG. 17. Observation by pump/probe spectroscopy of an electronic wave packet oscillation in an asymmetric coupled quantum well structure (Leo ef a/., 1991). As the electrons oscillate between the two wells, they modulate the transmission of the probe beam.

hdQ,,, = 3.2meV (Leo et al., 1991). The amplitude of the oscillations depends on the bias voltage, translating the proximity to resonance. Clearly, these are damped, showing that the electronic wave packet loses its coherence as it moves back and forth between the two QWs. Although the electron motion is well described by the single-particle model sketched earlier, it is necessary to take into account BCI between electron and hole (Bastard er a/., 1989) to reproduce the experimental interband transition energy (Fox et al., 1990). Since the early days of solid-state physics, it was argued that because of the k-space periodicity of the energy dispersion of carriers in a crystal, an electron subject to a constant electric field F would perform an oscillatory motion both in r-space and in k-space (Bloch, 1928; Zener, 1932). Such dynamics, called Bloch oscillations (BOs), were never observed in bulk

3 OPTICAL PROCESSES IN SEMICONDUCTORS

225

crystals because, to execute one BO, the electron would have to reach the edge of the Brillouin zone, thus gaining an energy on the order of the band width, i.e., a few electronvolts, without experiencing any scattering, as the analyses of Bloch (1928) and Zener (1932) assumed. A much more favorable situation for observing BOs is provided by semiconductor superlattices (SLs). Here one works with the envelope wavefunctions, the SL period d, and SL minibands rather than with the Bloch wavefunction, the lattice period, and the energy bands. This change of scale brings, among other things, the BO period zB0 = h/eFd into the picosecond range. In the presence of a static electric field, the absorption spectrum of SLs exhibits the so-called Wannier-Stark ladder (WSL) structure (Mendez et al., 1988; Voisin et al., 1988). This consists of evenly spaced transitions EN = E , + N A E , with N = 0, f 1, f2, ..., where A E = h/TBO = eEd, between the SL electronic states and a hole state that is localized owing to the large mass of the holes. Again, excitonic effects strongly modify the interband transition energies and must be accounted for properly (Digman and Sipe, 1990). In a certain sense, the WSL structure is the frequency-domain manifestation of the BO (Bastard and Ferreira, 1989). Nevertheless, the direct observation of BOs in the time domain remained a challenge until ultrafast time-resolved nonlinear optical spectroscopy techniques were exploited. The idea is rather similar to that discussed in the preceding paragraph. An ultrashort pulse whose spectrum covers several WSL transitions would create an electronic wave packet that would oscillate with the period zB0. This charge oscillation should be observable in an FWM experiment because it would modulate the interband polarization. A clear signature of the BOs that distinguishes them from other oscillations, such as lh-X/hh-X QBs, is that their period zBo = h/eFd depends on the applied field and is therefore tunable. This scenario was indeed applied, using SLs located in the intrinsic region of a p-i-n heterostructure (Feldman et al., 1992; Leo et al., 1992; 1993; Leisching et at., 1994). An example of BOs observed through TI F W M is shown in Fig. 18. The field dependence of the period is seen clearly. Strictly speaking, however, it is not clear that the features seen in the TI FWM correspond directly to an oscillation of the center of mass of the electronic wave packet, since a symmetric breathing mode of the envelope wavefunction also could modulate the TI FWM signal as well. Recently, an elegant experiment (Lyssenko et al., 1997) was able to actually directly measure the spatial amplitude of the electronic wave packet. The experiment is based on the observation that as the wave packet oscillates, it creates a field that superimposes itself on the constant applied field F, thus modulating the spectrally resolved FWM whose maxima experience shifts as At is varied. The magnitude of the oscillating dipole, and hence the amplitude of the center of mass motion, is directly related to these shifts. The center of mass

226

D. S. CHEMLA ,

"

-

.

I

"

'

I

FIG. 18. Observation of the BIoch oscillations in a superlattice structure by four-wave mixing for a set of applied electrostatic fields (Leo er a/., 1992). The electron oscillations modulate the four-wave mixing signal.

of the wave packet executes damped sinusoidal oscillations, as shown in Fig. 19. The amplitude is macroscopic, -14nm for the first oscillation, and follows, quite closely, the theory (Lyssenko et al., 1997).

VII. Fundamentals of Four-Particle Correlation Effects Involving Real Electron-Hole Pairs

So far I have been able to describe the main observations by accounting for Pauli blocking and two-particle correlation at the HF/RPA level. As mentioned in the introduction, coherent nonlinear optical processes involv-

227

3 OPTICAL PROCESSES IN SEMICONDUCTORS

Time (ps)

-

FIG. 19. Measurement of the amplitude of the wave packet center of mass motion during Bloch oscillations in a superlattice (Lyssenkoet a/., 1997). The maximum amplitude is 14nm.

ing bound biexcitons have been investigated extensively in bulk semiconductors in the nanosecond regime (Maruani and Chemla, 1981; Chemla and Maruani, 1982). Obviously, processes involving two excitons require a description accounting for four-particle correlations at least. Effects associated with bound states of biexcitons are easily identified in 11-VI and I-VII semiconductors because their binding energy AEX2= h(Rx, - UZ,) is large enough that the two-photon biexciton resonances, 2 0 z Rxl are well separated from the one-photon exciton resonances 0 = 0,.In 111-V materials, biexcitons have a very small binding energy AEx2 < 1 meV and were not expected to play an important role in nonlinear optics. In quantum confined structures, although AEx, z 1 to 3meV is enhanced (Miller and Kleinman, 1985), it remains on the order of the exciton linewidth in these inhomogeneously broadened systems. Thus it came rather as a surprise when oscillations at a frequency different from the Ih-X/hh-X splitting were observed in GaAs/AlGaAs QW structures through pump/probe (Bar-Ad and Bar-Joseph, 1992) and F W M (Lovering et al., 1992) experiments. An example of the exciton/biexciton oscillations seen in pump/probe experiments (Bar-Ad and Bar-Joseph, 1992) is shown in Fig. 20. The origin of these new features was correctly identified as due to the bound biexciton contribution, which appears in coherent processes because a two-photon transition (one g - from the probe and one ' 6 from the pump) directly

228

D. S. CHEMLA

-10

-5

0

5

10

time delay (ps) FIG. 20. Exciton-biexciton oscillation observed in a GaAs quantum well sample by pump/ probe technique using a o- probe and a u+ pump (Bar-Ad and Bar-Joseph, 1992).

connects the ground state to an X, state, no matter what the inhomogeneous broadening is. Phenomenological five-level models including the ground state Is), the two X i excitons, a bound biexciton, and unbound exciton pairs and accounting for inhomogeneous broadening were proposed to interpret the data (Finkelstein et a/., 1993). More systematic studies (Mayer et al., 1994; Kim et al., 1995; Mayer et al., 1995) using three-pulse FWM in a number of polarization configurations were able to separate the quantum beats between ( l h - X i ) and (hh-Xi) from those between these excitons and their bound (Ih-X*), and (hh-X'), states. Again, the FWM signals measured in these experiments have a very clear polarization selectivity: strong FWM signal for 11-polarization, weak signal for Ipolarization and cocircular o*/o*-polarization, and almost vanishing signal for countercircular o*/o'-polarization (Mayer et al., 1994; 1995; Denton et al., 1995). Interestingly, it was found that the phase of the Ih-X*-hh-X* quantum beats seen in I-polarization configuration exhibit a clear n shift as compared with those seen in /-polarization. Describing qualitatively the overall line shapes of these experiments required the extension of the phenomenological models to 10 levels! The correct interpretation of these experiments requires formalisms able to handle n-particle correlations, including many obviously important mechanisms such as screening, which are not described by the HF/RPA of

3 OPTICAL PROCESSES IN SEMICONDUCTORS

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the SBE (Binder and Koch, 1995; Spano and Mukamel, 1989; 1991; Dubovsky and Mukamel, 1991; Leegwater and Mukamel, 1992; Mukamel, 1994; Axt and Stahl, 1994; Victor et al., 1995; Maialle and Sham, 1994; Ostreich et af., 1995; Axt et af., 1995; Schiifer et af., 1996). In the continuum of almost free e-h pairs, one could use nonequilibrium Green’s functions and a second Born approximation with a satisfactory accuracy. However, in the domain of highly correlated e-h pairs, the Coulomb interaction must be accounted for consistently to arbitrary order. Several theoretical approaches have been proposed for achieving this goal. The first one was developed in the context of molecular systems and, because of this, has received little attention from the “semiconductor community,” although it is absolutely general (Spano and Mukamel, 1989; 1991; Dubovsky and Mukamel, 1991; Leegwater and Mukamel, 1992; Mukamel, 1994). A formalism that naturally extends the density-matrix approach of the SBEs and is able to account for high order correlation is called the dynamic controlled truncation scheme (DCTS) (Axt and Stahl, 1994). Other formalisms that proceed through diagrammatic techniques (Maille and Sham, 1994) or through the development of correlation functions in the basis of n-exciton eigenstates have been proposed recently (bstreich et al., 1995). I will base my discussion on the DCTS because it permits me to maintain continuity with previous sections. The DCTS consists of (1) writing the Heisenberg equations of motion for all relevant products of operators, (2) applying the fundamental Fermion anticommutation rules, and (3) taking the expectation values. This results in an infinite hierarchy of equations of motion coupling the n-particle and the (n + m)-particle correlation functions. The consistent truncation scheme is based on the fact that in nonlinear optics one is usually interested in a development in powers of the interaction Hamiltonian (Eq. 23) so that the electric field E(t) is the natural expansion parameter. When one wishes to describe the effects up to E(t)”, the system of coupled equations is truncated at this power and the terms O[E(t)”””] are neglected. This results in a closed system of equations that, in principle, can be solved exactly. The four-particle correlation functions that appear in the development of the kinetic equations beyond the two-particle correlation functions (Eq. 24) have the form (Axt and Stahl, 1994) N e - e = (2{2i2324)

Nh-h= (hfhih3h4)

and

N X = (2fhih32,) (49)

or

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D. S. CHEMLA

and similar ones observed by permutation of the indices or Hermitian conjugation of the e and h operators. Here, to simplify the notation, I have lumped all the quantum numbers that distinguish particles into a single index. The four-particle correlation functions represent coherent processes with a very intuitive interpretation. Those appearing in Eq. (49) can be interpreted, respectively, as an e-density/density correlation, an h-density/ density correlation, and an X-occupation, whereas those in Eq. (50) represent an e-screened lpair-emission, an h-screened lpair-emission, and a 2pair-emission. It is worth noting that they correspond to coherent processes, in contrast to the products of two-particle correlation functions that are deduced from them by applying an R P A and thus destroying the phase relation between the terms of the products. For example, the four-operator product appearing in P;k‘:&, C\C2P,h, represents the single process in which an e-h pair (e3, h,) and an electron (e2) are destroyed while an electron (el) is created, hence the interpretation of as an e-screened lpair-emission, i.e., (e3, h,)-recombination accompanied by the e2 + e, scattering. The intuitive picture carried by Eqs. (49) and (50) relates nicely to mechanisms that are usually considered in a more heuristic description of light-matter interaction; the DCTS formalism, however, is completely consistent. It is clear that as the order of the development increases, the number of correlation functions to consider becomes quickly unmanageable. It turns out, however, that it is possible to develop a systematic procedure for identifying all those which contribute at a given order (Victor el al., 1995). In the limit of third-order processes, r truncation, a number of factorization-summation relations valid O[E(tr’ j] can be demonstrated. They take x (Cjh4) O[E(t)‘], forms like (P::(l&,)* = (i3h;h:h4> = X,,(P\h;h$?i) showing that many coherent four-particie processes can -be expressed in terms of only two types of correlation functions: the lpair-transition, Peh = ($), the Zpair-transition, Bheh’,’, and their complex conjugates (Schafer et al., 1996). A physically meaningful expression for the excitonexciton correlation function Bhe’“e’appears naturally when operator products that have already been factorized in the SBE approximation are subtracted from the bare four-particle correlation function (with proper sign changes due to Fermi operator commutation rules). Then €Ihe’”,’takes the following form: Behe‘h= (Ph2h’) - (d)(2h’) + (Ch’)(2h) (Schafer et al., 1996) and has a straightforward interpretation. It characterizes the deviation from the H F/RPA meanfield theory. This underlying physics makes the coupled equations of motion of Peh and Bheh’e’much more transparent because they can be written in such a way that the first term in dFh/dt(,,, exactly reproduces the SBEs. This procedure shows that the DCTS includes, of course, the HF/RPA meanfield formalism (Schafer et al., 1996); furthermore, it is very useful in practice because it identifies the processes that are

c$&

1

-

1

+

3 OPTICAL PROCESSES IN SEMICONDUCTORS

231

or are not included in the SBEs. Discussing the details of the theory is beyond the scope of this chapter. Therefore, to get an insight into the mechanism relevant for the experiments mentioned in the preceding paragraph, I will use a generalization of the EPM (Eqs. 45 and 46) that, as we have seen, gives a good intuitive picture of the physics. The model proceeds along the same lines as EPM. If we consider only the hh -P e transition, we have to account for the two spin manifolds and introduce two effective polarizations P * , respectively, associated with the absorption of of photons and, in addition, an “effective four-particle correlation function” W that represents a two-photon (one o+ and one o-) bound biexciton transition. With these ingredients, one can derive from the full kinetic equations the coupled system:

and

In Eq. (51) Y is the effective BCI coupling, and therefore, the first line reproduces the SBE approximation (Eq. 45). All the other terms in Eqs. (51) and (52) originate from correlation effects beyond HF/RPA; YyFenis the effective parameter describing excitonic screening, YSf;This the corresponding exchange term, and Y x xaccounts I for the exciton-biexciton interaction. It is worth noting that the excitonic screening couples the two exciton spin manifolds and explains the coupling between P(t)- and P(t)+,whereas the excitonic exchange term has the same form as the BCI and can be lumped with it. The effective four-particle correlation function s(t)is driven by a product of two effective polarizations, P(t)- x P(t)+.The model readily explains how processes not included in the SBEs affect FWM experiments and govern their polarization selectivity (Axt et al., 1995; Schafer et al., 1996). In the f 3 ) limit, that is, O[E(t)’] solution, the polarizations in the a E(t). In the RHS of Eqs. (51) and (52) are the linear polarizations Pc1)*(t) configuration oflo*, either P(t)- or P(t)+ is zero, the RHS of Eq. (52) vanishes, and bound biexcitons are not created. All the other terms contrib-

232

D. S. CHEMLA

ute at the same level, and Eq. (51) has the same form as the SBE approximation (Eq. 45). In the ((-polarization configuration, the RHS of both Eqs. (51) and (52) are nonzero; therefore, bound biexcitons are created. The oscillations at frequency 2R, - R,, arise naturally, 9'(f)-is coupled to :Ip(t)' through VTY",and the FWM signal is large. Finally, in the 1-polarization configuration, the coefficient of V F y vanishes, the excitonbiexciton interaction contribution is dominant, but the signal is weaker. Other, apparently simple processes also imply correlation beyond more than two particles. This is the case, for example, for the dephasing induced by the presence of other charged carriers or other excitons. In Section V I mentioned that early FWM experiments, where e, h, or X populations were intentionally photogenerated, were described satisfactorily by using empirical density-dependent scattering rates y = yo + yini, with i = e, h, or X in analogy with the concept of collisional broadening of atomic physics (Schultheis ef al., 1985; Schultheis et al., 1986a; 1986b). It is easy to convince one's self that such density-dependent scattering rates imply processes beyond the third order. As mentioned in Section 111, when introducing a density-broadening parameter in the kinetic equations (Eqs. 13 and 14), terms of the form x pt appear, and since ne x nh x Ip,,12 + lpkI4 + O(ES)and P k = O(E), they contain contributions o(E'). Recently, the effects of collisional broadening have been reexamined carefully, their consequences on the FWM emission were clarified, and they were baptized excitation-induced dephasing (EID) (Wang et al., 1993; 1994). f t was found that even at low excitation densities, the exciton resonances experience a significant broadening. This is shown in Fig. 21, where the pump/probe DTS, measured near the X resonances of a 0.2-pm GaAs sample kept at low temperature, is presented. In this experiment, the pump generated Plyhx 3 x 10'' cm-3 excitons. The DTS is very well interpreted as the difference between Lorentzian resonances with the same strength but slightly different widths. FWM experiments using a prepulse to introduce a controlled amount of excitons long enough before the arrival of pump and probe pulses on the sample to be incoherent with the excitons involved in the FWM confirmed the earlier results. Moreover, these experiments have shown that as Npcohx 6 x 10'4cm-3 -+ 5 x l O l ' ~ m - ~the , FWM effivaried ciency was reduced by a factor -6, whereas the ratio SkWM/S+wM from 10 + 5. All these changes were found to be independent of the spin of the incoherent excitons created by the prepulses. Obviously, the Coulomb coupling between the two subsystems of excitons is responsible for the observations. Since screening in its various forms is not included in the SBEs, a description of EID requires using formalisms such as the DCTS. In fact, up to O[E(1)51, the discussion of the preceding paragraphs gives a pretty good

3 OPTICAL PROCESSESIN SEMICONDUCTORS

1.5075

1.5125

233

1.5175

Energy (eV) FIG.21. Differential transmissionspectra showing the broadening of the exciton resonance due to dephasing induced by n = 3 x 10”cn1-~photogenerated excitons (Wang et al., 1993). The DTS line shape is very well described as the difference between two Lorentzians. Inset: Linear absorption spectrum of the sample.

idea of the mechanisms involved. The exciton screening, i.e. the term cc V T Y ,introduces the channel that couples the X + and X - populations, and the ratio SbWM/S& >> 1 follows from Eq. (51). At the O [ E ( t ) 3 ]level, exciton screening does not affect the dephasing, and one has to look for an explanation beyond that order. A full and consistent theory of wave-mixing experiments, including a correct description of screening, is such a formidable task that it has not yet been attempted. In the experiments of Wang er al., (1994,1995), because the two exciton populations are completely incoherent and the densities are rather small, it is possible to use a less general scheme to explain the main trends (Hu er al., 1994). An effect of screening by the incoherent excitons is to renormalize the transition energies between the conduction and valence bands, (Eq. 28). In the conditions considered here, the self-energies CC,”can be estimated within the second Born approximation, in which all terms up to 0 [ 1 5 ( t ) ~ ] are included in the screened potential (Haug and Schmitt-Rink, 1984). The real and imaginary parts of C,, describe, respectively, a shift and a broadening of the single-particle levels. The exciton energy is very robust, however, because of the cancellation between band shift and binding energy (see Section V), and only the broadening remains. By developing the correspond-

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D. S. CHEMLA

+

+

ing parameter to the first order in the density y = yo yX(N?Coh NYh), the heuristic approach of Schultheis et al., (1985, 1986) is recovered and justified (Hu et al., 1994). In the context of the experiments of Wang et al., (1994, 1995). this introduces in the SBEs the desired X + and X - coupling, as in the DCTS, and accounts for the observed broadening of the exciton resonances. In a certain sense, EID provides new source terms for the nonlinear polarization whose effects appear in many nonlinear optical processes. This is indeed the case, and the coherent transients associated with the EID due to exciton-continuum scattering have been observed in GaAs by 15-fs short-pulse FWM experiments that excite the resonances as well as e-h pairs in the continuum (Wehner et d.,1996). I t was expected that the bound biexciton states would make a noticeable contribution in processes where two-photon transitions are active, and it was rather implicit that the continuum of unbound exciton pairs would play a minor role. Surprisingly, this is not the case. Recently, it was found that correlations in the continuum of X-X scattering states can have important, and even dominant, erects in regimes of distorted excitons at low density (Kner et al., 1997a; 1997b; Lovenich er al., 1997). A magnetic field B’(\Z applied to a semiconductor confines electrons and holes in the (2, j ) plane, inducing a 3D-to-1D transition for the density of states and strongly modifying the internal structure of the excitons. They experience a shrinkage, K in the (2,2;) plane and cc In(lB1) 11 5.The magnetic confinement is expected to have significant erects at fields (B(>> B,, where B, is the field strength at which the cyclotron radius equals a,, the ( B = 0) excitonic Bohr radius (Lerner and Lozovik, 1981a; 1981b; Stafford et al., 1990-1; Kovolev and Lieberman, 1992; 1994). For semiconductors, this regime can be explored easily, for example in GaAs B, z 3.4 with the further practical advantage of an adjustable confinement while studying the same volume in a single sample. This continuous tuning of the manybody interactions governing the nonlinear optical response provides a perfect laboratory for studying manybody interaction processes. Figure 22 shows the experimental TI FWM for different magnetic field strengths up to B = 10 T z 3B, in an optically thin (0.25 pm), high-quality GaAs layer with homogeneously broadened Ih-X and hh-X excitons ( y z 0.4meV), which are visible because of mechanical strain (Pollak and Cardona, 1968). The measurement was performed with cocircular polarization a-/a- to minimize the effect of bound biexcitons. The signal at B = 0 T shows an exponential decay superimposed on oscillations for At > 0, with a dephasing time T2 z 1.5 ps and oscillation period corresponding to the Ih-X/hh-X splitting. For At < 0, the signal is much smaller, and its rise time is -3OOfs. As B is increased, the At > 0 signal S;, changes only slightly, wherease the At < 0 signal S; changes drastically. Its magnitude increases significantly relative to S:,, and

m,

3 OPTICAL PROCESSES IN SEMICONDUCTORS

235

FIG.22. Time-integrated four-wave mixing signal measure with u - / u - polarization in a GaAs sample as a function of the applied magnetic field (Kner et al., 1997). As the magnetic field is increased, the At < 0 signal acquires a long and nonexponential profile that can be seen as far as 100 times the laser pulse duration. It signals four-particle correlations not accounted for in the time-dependent Hartree-Fock theory.

-

above B x B,, the rise time lengthens to 3ps while the profile becomes highly nonexponential with an unusual positive curvature. As the density is lowered to N x 5 x 1014cm-3,i.e., at an average exciton-exciton separation as large as d x lOu,, S;, can be seen as far as Ar x -lops, i.e., 100 times the pulse duration! Furthermore, it was confirmed that the oscillations are quantum beats and not polarization interference by inspection of the slope of the peak of the TR FWM versus At (Koch et ul., 1992) and, more

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D. S. CHEMLA

important, because it was found in the spectrally resolved FWM signal S,(u, At) that the Ih-X and the hh-X significantly exchange oscillator strength as Ar is varied. In fact, that exchange can be so strong that Ssp(o, At) could be completely dominated by the Ih. This is in contradiction to the SBEs, which predict an N1-X contribution about an order of magnitude smaller than that of the hh-X and constant relative contributions of the Ih-X and hh-X to the signal. Again, these observations require accounting for four-particle correlations. Before showing the result of full numerical simulation of the experiment within the DCTS formalism, let me again use the extension of the EPM to gain an intuitive insight into the physics. In general, 93 encompasses both bound biexcitonic states and unbound biexciton states. In the C T - / O configuration, the former are not active, and only the X-X scattering states need to be considered. In this case, it is easily found that 9’and 9 obey the coupled equation system:

and

where now W accounts only for the X-X scattering states, which are modeled as a single resonance at 2Q,, and *v;, lumps all the exciton-exciton interactions active in the a-/cr- configuration. One recognizes the EMP model of the SBEs (Eq. 45) in the left-hand side and first two terms of the RHS of Eq. (53). Equation (54) can be formally integrated and put in Eq. (53), giving a third source term:

due to exciton-exciton correlation (XXC). This new term is obviously of the same order as the PB and BCI contributions; however, it has a completely different, non-Markovian time dependence; i.e., it grows first as the integral of the square of the polarization before exhibiting an exponential decay. The origin of this “coherent” memory is easy to interpret. Clearly, for a third-order signal in the direction 2x2 -XI, P(t)* is generated by E(xl),

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3 OPTICAL PROCESSESIN SEMICONDUCTORS

whereas the integral over P(t')' comes from @,). Therefore, when the R, pulse arrives first, at At < 0 it generates a four-particle correlation cc 9(t')' which, because it corresponds to a two-photon transition, cannot emit light pulse arrives and triggers the emission and builds up as f&dt'. ..until the of the FWM signal. The TI FWM response of Eqs. (53) and (54) can be calculated easily. An example is shown in Fig. 23 for values of the parameters chosen to reproduce the experiment of Kner et al., (1997) and displaying the separate contributions arising from the PB, BCI, and XXC. Of course, only the general features are reproduced by this simple model, and many details, such as the Ih-X/hh-X beats, are not included. Neverthe; is reproduced and that it has the long less, one sees that the large S duration and nonexponential profile with the positive curvature for small A t
x,

-2.0

-1.5

-1.0

-0.5

0.0

05

1.0

1.5

2.0

Time Delay @s) FIG. 23. Calculation of the timaintegrated four-wave mixing (TI FWM)signal using the "effective" polarization and excitonexciton correlation showing the contrasted time dependence of the Pauli blocking (PSF), the bare Coulomb interaction, and the exciton-exciton correlation contributions (Kner et al, 1997).

238

-6

-4

-2

0

2

4

6

- P a d Blocking + BareCoalomb + Exeiton-Exdton correlation FIG. 24. Theoretical time-integrated four-wave mixing signal calculated for GaAs at R = IOT and low density using the ‘density controlled truncation scheme” formalism. The large At < 0 signal is due to the exciton-exciton correlation (Kner et d.,1997).

data. Their attenuation in the experiment is most likely due to excitondensity correlations that contribute beyond the coherent limit. The dashed curve is the result of the calculation without XXC. Here we see that the signal without XXC has a much faster rise time for A t < 0. Furthermore, the XXC contribution completely dominates the FWM; for Ar < 0, the signal without XXC is three orders of magnitude smaller than the signal with XXC, and even for A1 > 0, it is still an order of magnitude smaller. The theoretical spectrally resolved FW M signal calculated for different At values recovers the enhancement of the Ih-X signal and the exchange of oscillator strength between the Ih-X with hh-X with Ar. These quantum beats, not predicted by the HF/RPA, are due to the strong coupling between the Ih-X and the hh-X in the four-particle correlation functions because of the distortion of their relative-motion wavefunction and their large mass difference. Looking for an intuitive explanation for the enhancement of the XXC by the magnetic field, it is worth noting that the four-particle correlation function Behe‘h’, which includes all the X-X interactions, contains the X-X multipole interaction in the long-wavelength limit (very low density). Clearly, in high magnetic fields, excitons are squeezed in the (2,7 ) plane and develop large

3 OPTICAL PROCESSES IN SEMICONDUCTORS

239

quadrupole moments that allow them to interact via a long-range quadrupole-quadrupole interaction. A t this point it is useful to comment on the reason why the excitonexciton correlation function Behe’h’naturally appears in the description of effects that are beyond the HF/RPA level of the SBEs. As mentioned in Sections IV and V, at that level of approximation one obtains a mean-field theory where the order parameters are the pair amplitude Peh and the electron and hole occupation numbers n, and n h . It is thus natural that the effects not included in that theory involve the exciton-exciton correlation function Behe‘h’= (2halk) - (i?h)(alk) (i?h’)(alh), since it measures the difference between the bare four-particle correlation function (;hi?%’) and the two products of two-particle correlation functions (?h)(Ph’) and (&’)(?’h) derived from it in the HF/RPA factorization. In the experiments described earlier, the time delays that are probed are short compared with the mean free time for X-X scattering. Thus not enough scattering events happen over the time span of one experiment for each X to interact with a substantial fraction of its neighbors, i.e., for the HF/RPA mean-field conditions to be established. These experiments, therefore, access the new regime where the fluctuations in X-X scattering induce large fluctuations of the HF/RPA mean-field order parameters. I will come back to the generality of this comment in my conclusion. This is not the end of the story. Nonlinear optical effects are extremely sensitive to the interactions between elementary excitations. They provide direct information on processes that are inaccessible to other spectroscopic techniques. Therefore, it is most likely that investigations of high-order manybody effects through nonlinear optical spectroscopy are going to be an important direction of research in condensed-matter physics. Already in the experiments described earlier there are indications that mechanisms beyond x(3) are active (Kner et al., 1997), and recently, there have been reports of unambiguous observation of x(’) and x”) processes (Wehner et al., 1997). Although the general framework for describing these effects exists in principle, the detailed theory is far from being developed, and one can anticipate surprises.

+

VIII. Dynamics in the Quantum Kinetics Regime We have seen in Section 111 that the most nonclassical dynamics regime occurs at very early times after e-h pairs are created. The time scale of this regime is determined by the period of the elementary excitations, plasmons, and phonons for semiconductors. Therefore, in the first few tens of femtoseconds after excitation, one expects to see new features in the nonlinear

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optical response of these materials. These can be used for investigating this poorly understood thermodynamic regime. This is the topic of this section; 1 shall discuss in turn the effects of carrier-carrier scattering and those of carrier-phonon scattering. Evidence for non-Markovian behavior was found in experiments where both the amplitude and the phase of an FWM signal were measured (Bigot er a!., 1993; Chemla er al., 1994-11;Chemla and Bigot, 1995). Through a combination of interferometric, time-resolved, and frequency-resolved measurements, a “time-energy” picture of the process was developed. An example of such a study is shown in Fig. 25. The GaAs QW sample is weakly excited, N z 3 x 109cm-2, just below the hh-X resonance. The left curve gives the logarithm of S-,.,(Ar); the arrows mark the Ar at which the spectrally resolved signal Sps(o, Ar) shown in the central panel and Awr), the phase (relative to that of the reference laser) shown in the right panels, were obtained. For comparison, the laser spectrum is depicted as a dotted line in the Ar = 0 graph of Sps(o, Ar). For Ar = -8Ofs, the emission spectrum is essentially at the hh-X, and Awr) shows an almost linear slope corresponding to a constant emission frequency A w l ) = (o- w,)r = 0. For At = 0, &@) has an asymmetric profile with a low-energy tail extending well into the laser spectrum, indicating that the instantaneous frequency of the emission is chirped. Correspondingly, Awr) exhibits a linear part first, but after about 250fs the slope changes, and after 350fs it flattens. At Ar = 160fs, Sps(u,Ar) has two separate contributions, one close to the hh-X and the other approximately following the laser spectrum. Awr) takes a more pronounced S shape, indicating that the emission is first centered at the laser, w z w,, then shifts at the hh-X, o z R,,,,, and then moves back again to the laser, o z w,. The experiments were modeled by an SBE theory, with screening treated in the static single-plasmon pole approximation and with dephasing accounted for by a constant rate. The calculated &,(At) (dashed curves) are unable to account for the line shape. A “frequency dependent” rate T(w) would give better agreement and would correspond to a memory kernel r(t - r’), as seen in Section 111. The phase dynamic within one ultrashort pulse is governed by events occurring within a few optical cycles; during such short times, the elementary excitations only experience a few “collisions”; thus the phase cannot randomize and memory effects become apparent. Although the discrepancies between experiments and the SBE theory were traced back to the approximations used, the exact origin of the mechanisms at work in dpJdtl,,,,, were not precisely identified in Chemla and Bigot (1995). More recently, experiments have been designed specifically to investigate the non-Markovian regime, and theoretical simulations have been developed

241

3 OPTICAL PROCESSES IN SEMICONDUCTORS

Ak160fs -180

At (f8)

I

380 .mev

1.483ev

0

105

210

FIG.25. Dynamics of the coherent FWM emission versus time delay At. Left curve: Logarithm of the time-integrated FWM signal intensity versus At. Central panel: FWM power spectrum for three time delays between -80 and 16Ofs, as indicated by the arrows on the left curve. The solid curves are the experimental results and the dashed curves are the theoretical results of the semiconductor Bloch equations. For comparison, the laser spectrum is depicted as a dotted line in the At = 0 graph of the central panel. It is tuned slightly below the hh-exciton resonance. Right panels: Corresponding phase difference with the reference laser (Bar-Ad and Bar-Joseph, 1991).

to interpret them. In the work of Bar-Ad et al., (1996), the authors argued that since the Liouville equation (Eq. 26) relates A and aA/at, the simultaneous determination of both the DTS (ATIT) and its derivative with respect to At, G(DTS) = a(AT/T)/aAt versus w, and At would put very strict restrictions on any theory invoked to explain the data (Bar-Ad et al., 1996). Their experiments were performed on GaAs using a pump/probe technique, with independently adjustable pump and probe durations of 30 -,1OOfs. In experiments with rather long, ( 270 fs) pulses and moderate density, the DTS exhibits a spectral hole slightly red shifted relative to the pump spectum, in agreement with previous reports (see Figs. 9 and 10) (Knox et al., 1986; Foing et al., 1992). When pulses much shorter than the natural time scales TLoand Tplare used, no spectral hole is seen in the DTS, which is featureless and extends from below the pump central frequency all the way to the exciton edge. More important, for very short At smaller than the pump duration, the G(DTS) shows a uniform positive growth shifted toward

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the exciton that reverses and changes sign immediately at the end of the pump pulse [Fig. 26(a)] (Bar-Ad et al., 1996). This is indicative of generation in the medium, during the pump pulse, of a polarization out of phase with the probe field over a broad range of energy below the pump and of a sudden change in phase when the pump pulse ends. Interpretation of the experiments was attempted by calculating the DTS and G(DTS) using a four-band version of the SBE within the Boltzmann kinetics relaxation time approximation. The first attempts, consistent with Boltzmann kinetics, considered only dephasing and assumed no population relaxation. It was possible to qualitatively reproduce the DTS line shape with a value T, z 200fs that agrees with the time scale seen experimentally. The calculated G(DTS), however, presents qualitative discrepancies with experiments [Fig. 26(b)]]. It can reproduce neither the shift toward the band edge during carrier generation nor the negative G(DTS) seen close to the laser center frequency immediately after the pump pulse is over [Fig. 26(a)]. By including a population relaxation toward a Maxwell-Boltzmann distribution with the same instantaneous number of carriers and total energy as that generated by the pump pulse, it was possible to get better agreement with the experiments [see Fig. 26(c)]. However, this occurs only for unphysical population relaxation times much shorter than the dephasing time TI % 36fs << T2 and, furthermore, too short to be compatible with theories based on Boltzmann kinetics (Goodnick and Lugli, 1988; Collet, 1993). Any attempts to salvage Boltzmann kinetics by using more complicated models for dephasing, such as EID, (Hu er al., 1994) failed to remove the unphysical results: (1) T2 >> T,, and (2) TI << 2n/RLo and << 2n/RPl.The features observed are, however, consistent with quantum kinetics theories (see Fig. 6). In the experiments of Bar-Ad et al., (1996), both electrons and holes are generated, thus complicating the interpretation. An elegant technique allowing one to follow the evolution of the electron alone was implemented by Camescasse er al., (1996). In these pump/ptobe experiments, the 130-fs pump excites the hh --* e and lh -,e transition, but the 30-fs probe was tuned to the spin-orbit split-off transition. In the case of CaAs, the spin-orbit splitting is large enough, -340meV, that no holes are excited by the pump at that transition, and the probe measures only the effects of the electrons. Although very small spectral holes are seen in the DTS at the hh -+ e and Ih -,e thresholds for very early time delay, in this case, again, the DTS is immediately very broad and becomes featureless even before the end of the pump pulse (Camescasse et al., 1996). The DTS line shape and its evolution versus Ar are very similar to those reported in Bar-Ad er a/., (1996). They were calculated from the SBEs with Coulomb quantum kinetic scattering terms of the type of Eq. (20). The agreement with the experiment is remarkable; most of the features observed experimentally are qualitatively reproduced, including the overall line shape and the

-

+

243

3 OPTICAL PROCESES IN SEMICONDUCTORS

I

1.40

-20

I

-1 0

I

I

1.45

1.55

1.50

0

10

20

FIG. 26. Comparison between the experimental G(DTS), derivative of the differential transmission spectrum with respect of the time delay, and the best simulations using the semiconductor Bloch equations within the relaxation time approximation. (a) Experimental results for 304s pump and probe pulses on a 1-pm GaAs sample. (b) Theory with T2= 200fs and no population relaxation. (c) Theory with & = 200fs and a population relaxation 7,= 36fs toward a Maxwell-Boltzmann distribution with the same instantaneous number of carriers and total energy as that generated by the pump pulse (Bar-Ad et al., 1996).

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disappearance of the spectral holes for At 2 80fs. As mentioned in Section H I , because the excitation spectrum of a plasma is gapless, carrier-carrier scattering tends to produce broad and featureless distributions. Furthermore, the experiments and theories discussed earlier show that in the quantum kinetics regime, carrier-carrier interactions almost instantaneously scatter the carriers out of the energy window in which they were created, resulting in a distribution much broader than that of the pulse that generates them and even broader than that predicted by Boltzmann kinetics (El Sayed and Haug, 1992; El Sayed et al., 1992; Tran Troi and Haug, 1993; El Sayed et af., 1994-1; 1994). The other scattering processes that can produce memory effects are due to the carrier-phonon interaction. We have seen that in polar semiconductors this interaction is dominated by LO-phonon scattering, which has a well-defined and single frequency and forms a “single-mode reservoir,” as mentioned in Section 111. Very strong coupling of the exciton to LOphonons was observed in 11-VI nanocrystals (Roussignal et al., 1989; Schoenlein et al., 1993; Mittleman et al., 1994). In these systems, however, the electronic excitations form discrete levels because of quantum confinement in all directions, and the interaction with vibrations can be analyzed as for molecules (Schoenlein et al., 1993; Mittleman et al., 1994). For structures of higher dimensionality, the electronic levels form bands, and LO-phonon scattering results in intraband transitions normally associated with irreversible processes and dissipation. Non-Markovian effects due to LO-phonon intraband transitions were observed recently in FWM experiments, where bulk GaAs was resonantly excited by -14-fs pulses, much shorter than the LO-phonon period. As shown in Fig. 27, the TI FWM signal &,(At) presents a strong oscillatory modulation superimposed on the usual exponential decay. The period, 98 fs, corresponds to the separation between two conduction band states coupled by one LO-phonon, i.e., (1 + m,/m,)RLo (Bhnyai et al., 1995; Steinbach et al., 1996). The amplitude of the oscillations decreases when the excitation density increases. Surprisingly, these features are not reflected in the spectrally resolved signal Ssp(w,At), which at each At consists of a single line without phonon sidebands (Banyai et al., 1995; Steinbach et al., 1996). This implies that the electrons excited by the first pulse interact with the LO-phonons, which in turn affect, at a later time, electrons involved in the transitions of the second pulse, i.e., the kind of memory of the electron subsystem interacting with the LO-phonon thermal bath that we discussed in Section 111. An analysis based on an SBE description of the FWM but with a J p J d t described by quantum kinetics scattering integrals for the interaction for the electron and phonon subsystems (Haug, 1992; Banyai et al., 1992), equivalent to that of Kuznetsov (1991) [see Eq. (17)], was able to

-.,

3 OPTICAL PROCESSES INSEMICONDUCTORS

245

100 * 00

-

0 w I-

U CT

1.00

LL

tf

0.10

0.01

200 TIME DELAY (f s)

0

100

FIG. 27. Time-integrated four-wave mixing signal measured in a GaAs sample at T = 77 K with -14-fs pulses and for three carrier densities n,, % 1.2, 1.9, 6.3 x 1016cm-3.The dashed curve, marked AC, shows the laser autocorrelation.The dots show the results of the quantum kinetics theory (Binyai et al., 1995).

explain quite nicely the data (Banyai et al., 1995; Steinbach et al., 1996). In particular, the period of the oscillation was related to beats of interband transitions whose electronic states are coupled by an LO-phonon. Phonon oscillations are also expected to appear in the DTS of pump/ probe experiments. In the early investigations, where rather large excitation densities were used (Lin et al., 1987; Schoenlein et al., 1987), the DTS was dominated by carrier-carrier scattering, and phonon replicas were not observed. With improvement of the laser and detection techniques, the topic has been revisited successfully recently (Leitmstorfer et al., 1996; Furst et al., 1997). Using a combination of low excitation density, -8 x 1014cm-3, ultrashort probe pulses, 25 fs, and circular polarization selection rules to detect only the carriers generated in the hh 4e transition, several phonon replicas were observed in GaAs (Fiirst et al., 1997). Broad features are seen at very short At ( I 80 fs) due to the energy uncertainty relation, but over times on the order of the LO-phonon period, the phonon replicas start to appear superimposed on a broad background. Their width narrows and becomes on the order of that of the laser at later times. In fact, it seems that

-

246

D. S. CHEMLA

the successive replicas appear one after the other and are roughly separated in time by an LO-phonon period. This indicates a succession of quantum interferences whose time scale is related to the internal period of the “thermal reservoir,” as discussed in Section 111. It turns out that an analytical solution for the electron-phonon quantum kinetic equation for a one-dimensional system was found recently (Meden et a/., 1996). Using this model to calculate the scattering integrals of an SBE description of the pump/probe experiments, theoretical DTSs in remarkable agreement with the experimental ones were obtained (Furst et af., 1997). To finish this section I want to discuss some important and recent developments. In the preceding paragraphs I have shown that what is usually called an “irreversible” process is, in fact, a quantum mechanical interaction with an oscillatory behavior that couples a degree of freedom of a subsystem with the numerous ones of a thermal reservoir. The process becomes really “irreversible” only after several oscillations. Therefore, it should be possible to reuerse or enhance, after it has already started, an interaction process that would become irreversible if the subsystem and the reservoir were left to themselves. Following this argument and the narrative of the preceding paragraphs, such a situation could be implemented by replacing one of the pulses of a canonical FWM configuration by a pair of pulses whose relative phase is properly chosen. This has been demonstrated in a recent FWM experiment (Wehner et af., 1998) where two phase-locked pulses, 1 and l‘, propagating in the direction 2,and separated by the time delay Ar = t , - 2 , . interact with a pulse propagating along delayed by Ar,,. = f, - t , . in a GaAs sample at T = 77 K. The F W M signal is detected, as usual, in the direction 21, -XI, so in the x‘3) regime the pair of phase-locked pulses enter linearly polarized. The 15-fs pulses are tuned to the band gap; they are all linearly polarized and have approximately the same intensity. The relative phase between pulse 1 and pulse 1’ is defined to better than 0.1 fs. If only onex, pulse is present, this is the same experiment as in Banyai et al., (1995), and the same results are reproduced. With the pair of phase-locked pulses, the ST,(Atz1.) shown in Fig. 28 exhibits remarkable features as At, is varied over a range corresponding approximately to one optical cycle. Depending on the time delay between the phase-locked pulses, the phonon oscillation can disappear completely (e.g., around A t , = - 43.64fs) or can be very pronounced (e.g., around Ar, = - 42.38fs). The detailed theory of these experiments is not yet available, but model calculations, able to reproduce the main trends, confirm this interpretation (Wehner et al., 1998). These experimental results clearly demonstrate that it is indeed possible to exploit the techniques of ultrafast nonlinear optics to manipulate the so-called irreversible scattering processes.

r,

,,.

,,

,,

247

3 OPTICAL PROCESSES IN SEMICONDUCTORS t

I

I

I

-100 0 100 200 300

TIME DELAY t21* (fs) FIG. 28. Coherent control of phonon scattering processes. Line shape of the time-integrated and four-wave mixing signal as a function of the time delay t 2 , , between the pulses along for a series of fixed time delays At,,. between two phase-locked pulses. The range of At,,., t , , , = -43.64 (top)-+-41.11fs (bottom), corresponds approximately to one optical cycle. It was scanned in steps of 0.21 fs. The GaAs sample is at T = 77 K the excited carrier density is nr.k z 3.6 x 1015m-3. Inset: Laser (dotted line) and TI FWM (solid line) spectra at zero time delay. The phonon oscillations are absent for t , , . = -43.64fs and pronounced for t , , , = -42.38fs (Wehner et al., 1998).

r,

z,

The quantum kinetics regime is still poorly understood. In particular, there are still outstanding questions about how to connect quantum kinetics and Boltzmann kinetics theories. Currently, our interpretations are based on an SBE description of 8pk/8tlcoh,augmented by 8pk/8tlScatt calculated using scattering integrals that rely on mechanisms beyond the HF/RPA level of the SBEs. One may look for a more consistent formalism, which may be derived from a treatment such as the DCTS that is, in principle, exact. Nevertheless, ultrashort-pulse time-resolved experiments are now able to explore regimes that were not accessible previously and will continue to provide novel information on the very short time dynamics of manybody interactions.

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IX. Conclusion As I conclude this chapter, it may be useful to reflect on the recent developments in the area of time-resolved nonlinear optical spectroscopy of semiconductors that I have just reviewed and to compare them with parallel advances in other areas of condensed-matter physics. In general, one can say that observing/describingmanybody effects has been a driving force for this whole field of physics. However, in most of the other subfields (quantum transport, superconductivity, quantum Hall effect, etc.), researchers are interested in understanding how degrees of freedom lock together as the energy scale is lowered. Correspondingly, they concentrate on the lowenergy elementary excitations and the “long time” scales. In this regime, one can probe the formation of “order parameters” and the establishment of mean fields. Several aspects of time-resolved nonlinear optical spectroscopy of semiconductors can be contrasted with this approach. First, optical processes with phonon energies close to the band gap of semiconductors correspond to the creation of elementary excitations whose dynamics evolve significantly on short time and short length scales. Therefore, what can be explored with ultrafast optical techniques is the new regime where the fluctuations of the “order parameters” become important and the mean-field pictures break down. Second, it is clear that the manybody mechanisms that are responsible for the formation of the quasi-particles seen in the linear regime are also responsible for the interactions among these quasi-particles that are seen in the nonlinear regime. However, what was discovered in the last decade is that these various interactions have specific dynamics with specific time scales and that they are strongly and differently affected by quantum confinement. Thus, as already noted, time-resolved nonlinear optical spectroscopy in quantum-confined systems provides information on manybody interactions that is inaccessible to conventional spectroscopic techniques. Finally, electronic states in semiconductors are well described by the effective mass and mean-field approximations, and their basic physics is well understood. Thus they constitute an almost ideal ground for testing advanced manybody theories. This, combined with important experimental advances, has placed the field of time-resolved nonlinear optical spectroscopy of semiconductors in an exceptional situation. By exploiting ultrafast time-resolved techniques under quantum confinement, it has been possible to design experiments for investigating manybody processes, in almost perfect samples, with an unprecedented flexibility and sensibility. This has provided theorists with a wealth of reliable and novel experimental data that motivated them to develop very refined descriptions of subtle phenomena and enormously further our understanding of manybody systems. It is not

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a big stretch of imagination to predict that the same approach will be applied soon to other outstanding problems of condensed-matter physics. In this chapter I have tried to give an overview of the spectacular recent progresses made in the time-resolved spectroscopy of semiconductors. Because of space restrictions, I had to concentrate on the conceptual and fundamental aspects of the subject and leave apart a number of interesting areas. Concerning the material systems, I have not covered the II-VI and I-VII compounds, mostly because the fundamental physics is quite similar to that of the III-V compounds, the main differences being due to the magnitude of some parameters such as the electron-phonon coupling or the binding energies of the exciton complexes. This, nevertheless, can affect significantly the carrier relaxation and correlation (Nurmikko and Gunshor, 1995; Gunshor and Nurmikko, 1996; Nurmikko, 1997). The magnetic semiconductors present, in addition, some extremely important and exciting spin-related effects that form a separate topic in their own right (Awschalom and Samarth, 1993; Cundiff et al., 1996; Crooker et al., 1997). The III-V nitride family promises to have a great impact on photonics and electonics in the next decade. Although the material quality has greatly improved recently, more progress must be made in defect control, but there is no doubt that the III-V nitrides will soon attract much attention in the area of nonlinear spectroscopy. In terms of elementary excitations, magnetoexcitons in quantum well structures (Stark el al., 1990; 1992a; 1992b; 1993; SchmittRink et al., 1991) and magnetically induced Fano resonances (Siegner et al., 1996a; 1996b) have complex and very interesting ultrafast dynamics that I have not reviewed. Finally, I have also left aside a number of applications, e.g., the generation and use of THz radiation from heterostructures excited by ultrashort laser pulses (Nuss and Orenstein, 1998) or the dynamics of microcavities where the number of photon modes interacting with the e-h pairs can be controlled (Khitrova et al., 1998). Covering all these topics would have required at least doubling the size of this chapter.

ACKNOWLEDGMENTS

It is a pleasure to acknowledge many very fruitful discussions with S. Schmitt-Rink, W.Schafer, S. Mukamel, I. Perakis, L. Sham, M. Wegener, and A. Stahl. I would like also to thank D.-H. Lee for his critical comments of the manuscript. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Division of Material Sciences of the U.S. Department of Energy, under Contract No. DE-AC03-76SF00098.

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LISTOF ABBREVIATIONSAND ACRONYMS AC AC FWM BCS BCI BG R BO CQK

cw

DCTS DOS DTS EID EMA EPM FES

autocorrelation autocorrelation four-wave mixing Bardeen Cooper Schockley bare Coulomb interaction band-gap renormalization Bloch oscillation Coulomb quantum kinetics continuous wave dynamic controlled truncation scheme density of states differential transmission spectrum excitation-inducing dephasing effective mass approximation effective polarization model Fermi edge singularity

FWM HF/RPA hh ih OSE PB PI PSF QB QW RWA SBE TI FWM TR FWM SL WSL

xxc

four-wave mixing Hartree-Fock/random phase approximation heavy-hole light-hole optical Stark effect Pauli blacking polarization interference phase space filling quantum beat quantum well rotating wave approximation semiconductor Bloch equation time-integrated four-wave mixing time resolved four-wave mixing superlattice Wannier-Stark ladder exciton-exciton correlation

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SEMICONDUCTORSAND SEMIMETALS VOL. 58

CHAPTER4

Optical Nonlinearities in the Transparency Region of Bulk Semiconductors Mansoor Sheik-Bahae DEPARIMEN~ ff PHYSKS AND A s r ~ c " U m m w NEW Mwm ALBUQOPIQUE, NEW MEXICO

Eric W Van Stryland Cmm R)R R W C H rn FDucATaN IN OPTICS AND Lnsw (CREOL) U m m f f CeMRM FLORIDA

.

Oluurm FLO~LVA

I. INTRODUCTION

............................

11. BACKGROUND. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Nonlinear Absorption and Refracrion . . . . . . . . . . . . . . . . . . 2. Nonlinear Polarization and the Definitions of Nonlinear Coeflcients . . . . . OF FWUND-ELECTRONIC NONLINEARITIES:TWO-BANDMODEL . . . . . 111. THEORY 1. Nondegenerate Nonlinear Absorption . . . . . . . . . . . . . . . . . . 2. Nondegenerate Nonlinear Refraction . . . . . . . . . . . . . . . . . . 3. Polarization Dependence and Anisotropy of x'3' . . . . . . . . . . . . . IV . BOUND-ELECTRONICOPTICAL NONLINEAR~ESIN ACTIVESEMICONDUC~ORS. . . V . FFSE-CARRIER NONLINEARITIES ......................

VI . EXPERIMENTALMETHODS . . . . . . . . . . . . . . . . . . . . . . . . 1. Transmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Beam Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Excite-Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Four-Wave Mixing . . . . . . . . . . . . . . . . . . . . . . . . . 5. interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.Z-Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Excite-Probe 2-Scan . . . . . . . . . . . . . . . . . . . . . . . . . 8. Femrosecond Continuum Probe . . . . . . . . . . . . . . . . . . . . 9. interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . APPLICATIONS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Ultrafast All-Optical Switching Using Bound-Electronic Nonlinearities . . . 2. Optical Limiting . . . . . . . . . . . . . . . . . . . . . . . . . . . Vlli . CONCLUSION ............................. LIST OF ABBREVIATIONS AND ACRONYMS . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

258 259 259 261 271 271 277 283 284 287 293 294 294 296 291 299 300 302 306 307 307 308 309 311 313 313

List of Abbreviations and Acronyms can be located preceding the references for this chapter .

257 Copyright 0 1999 by Academic Press All rights of reproduclion in any form reserved. ISBN 0- 1 2 - 7 9 167-4 ISSN 0080-8784199 530.00

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STRYLAND

I. Introduction The nonlinear optical properties of semiconductors are among the first studied (Braunstein and Ockman, 1964) and continue to be extensively investigated (Haug, 1988; Miller et al., 1981a;Jain and Klein, 1983) and used for a variety of applications (e.g., optical switching (Stegeman and Wright, 1990) and short pulse production (Keller et al., 1996; Kaetner et al., 1995) (See also Vol. 59, Chap. 4). Some of the largest nonlinearities ever reported have been in semiconductors (Miller and Duncan, 1987; Hill et al., 1982) and involve near-gap excitation. However, these resonant nonlinearities, by their nature, involve significant linear absorption (see Chap. 1 in this volume and Chap. 5 in Vol. 59), which is undesirable in many applications. In this chapter we concentrate on the nonlinear response in the transparency range of semiconductors, i.e., for photon energies far enough below the bandgap energy E,, that bound-electronic nonlinearities either dominate the nonlinear response or are responsible for initiating free-carrier nonlinearities (e.g., two-photon absorption-created carrier nonlinearities). The bound-electronic nonlinearities of two-photon absorption (2PA) and the optical Kerr effect are the primary nonlinearities of interest. The nonlinear optical behavior in the transparency region of solids due to the anharmonic response of bound valence electrons has been studied extensively in the past (Adair et al., 1987, 1989; Akhmanov et al., 1968; Flytzanis, 1975). Nonlinear refraction associated with this process is known as the bound-electronic Kerr efect. It is described by a change of refractive index An = n , l , where 1 is the light irradiance (W/cm2) and n2(cm2/W) is the optical Kerr coefficient of the solid. This type of nonlinearity results from virtual intermediate transitions (Boyd, 1992) as opposed to real intermediate transitions that occur in resonant (electron-hole plasma) nonlinearities. In the language of quantum mechanics, a virtual carrier lifetime can be defined from the uncertainty principle as l/[co-cogl. Here, w is the optical frequency, and a,, = E,/h, where E, is the bandgap energy of the solid, and h is Planck’s constant. This equality means that in the transparency region where W K o,,the response time is very fast (<< 10- I4s) and can be regarded as essentially instantaneous. This ultrafast response time has been exploited in applications such as soliton propagation in glass fibers (Agrawal, 1989) and in the generation of femtosecond pulses in solid-state lasers (Kerr lens mode locking) (Spence et a/., 1991). Another significant application is the development of ultrafast all-optical-switching (AOS) devices (Gibbs, 1985). Although much progress has been made in this area, development of a practical switch has been hindered by the relatively small magnitude of bound-

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electronic nonlinearities. The AOS issues are discussed in more detail in Section VII.1. The organization of this chapter is as follows: The next section (Section 11) is intended to provide an elementary background in nonlinear optics and familiarize the reader with the basic definitions and relations pertaining to the ultrafast third-order nonlinear response ( x ' ~ ) in ) materials. Section 111 contains a simple yet general semiclassical theory describing nonlinear absorption (NLA) and nonlinear refraction (NLR) in semiconductors. The key insight of this theory is the use of a Kramers-Kronig transformation to unite the refractive (Kerr effect) and absorptive (e.g., two-photon absorption) components of the bound-electronic nonlinearities. A simple twoparabolic-band model describes the band structure of the solid. The simplicity of this model allows for great generality, making it applicable not only to semiconductors but to large-gap optical solids as well. The results of the extension of this theory to include active semiconductors are given in Section IV. In Section V, cumulative (slow) nonlinearities due to generation of free carriers are discussed. Section VI gives a brief description of experimental techniques including Z-scan (Sheik-Bahae et al., 1989, 1990b), wave-mixing schemes (Adair et al., 1989), and interferometric methods (LeGasse et al., 1990), while Section VII discusses a number of potential applications, namely, all-optical-switchingand optical limiting. Finally, the conclusion of this chapter is presented in Section VIII.

II. Background 1. NONLINEAR ABSORPTIONAND REFRACTION The processes of nonlinear absorption (NLA) and nonlinear refraction (NLR) in materials, in the most general case, can be considered as the interaction of two light beams having distinct frequencies (0,and cob) in a nonlinear medium. In such an interaction, the two beams can alter each other's phase (NLR) or amplitude (NLA), the latter process requiring certain energy resonances. Note that in this type of wave mixing no new frequencies are generated. The preceding condition, known as the nondegenerate interaction, is the general case of the simpler degenerate situation where the two beams have the same frequency (0, = cob). Most convenient experimental arrangements, however, involve an even simpler degenerate geometry where both beams have the same vector as well as frequency. In this case, which is also known as nonlinear self-action, a single beam alters its own phase and/or amplitude through propagation in a nonlinear medium.

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In the characterization of a nonlinear material, one determines the nonlinear change of refractive index (An) and change of absorption coefficient (Aa) of a material. Relations for these changes are

and

where I , and I , are the irradiances of the two beams. Here, n, and a, refer to the nonlinear refractive index and nonlinear absorption coefficients, respectively. For photon energies h o i satisfying E g / 2 < h(w, + c o b ) < EB, a, accounts for 2PA and is often denoted by fi. Note that without loss of generality, we assume that the measurement is performed on beam 1, while beam 2 acts as an excitation source only. The first terms on the right-hand side of the preceding equations correspond to self-action (ie., single-beam experiments). The second terms correspond to the case of an excite-probe experiment provided that the two beams are distinguishable either by frequency and/or wavevector. The factor of 2 in front of the second term is a consequenceof this distinguishability, since a higher degree of permutation is allowed in the nonlinear interaction process (Boyd, 1992). This stronger nondegenerate response is sometimes referred to as weak-wave retardation (Van Stryland et af., 1982). While most reported measurements and applications involve degenerate self-action processes, in the theoretical treatment presented in this chapter we will consider the more general nondegenerate case while keeping in mind that the degenerate coefficients are merely the limit of the nondegenerate ones for w, = 0,. Aside from the generality argument, the reason for choosing this theoretical approach becomes more apparent when we discuss the use of Kramers-Kronig dispersion relations in relating the NLR and NLA. The nonlinear optical interactions presented in this chapter will be treated in two separate steps. The macroscopic propagation process (i.e., Maxwell equations) will be considered independently from the microscopic interaction that concerns the various mechanisms in the nonlinear response of the system. The propagation effects will be introduced first with the assumption that the nonlinear coefficients and their frequency dependence are known. The subsequent section will dwell on the microscopic calculation of the nonlinear coefficients using a simple, semiclassical two-band theory for semiconductors.

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2. NONLINEAR POLARIZATION AND THE DEFINITIONS OF NONLINEAR COEFFICIENTS a. ‘Ihird-Order Nonlinearities As defined in most nonlinear optics texts, the total material polarization

P that drives the wave equation for the electric field E is (ignoring nonlocality) (Boyd, 1992)

where R(”)is defined as the nth-order, time-dependent response function or time-dependent susceptibility. The subscripts are polarization indices indicating, in general, the tensorial nature of the interactions. The summation over the various indicesj, k, 1,. .. is implied for the various tensor elements of R(”). Ignoring the second-order effects [i.e., the second term on the RHS of Eq. (3)] corresponding to wave mixing and electro-optic processes in noncentrosymmetric materials, we focus our attention on the third-order nonlinearities. The third-order response is the first term that can directly lead to NLA or N L R . Upon Fourier transformation, we obtain

where 6 is the Dirac delta function. Here the nth-order susceptibility is defined as the Fourier transform of the nth-order response function:

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MANSOORSHEIK-BAHAE AND ERICW.VAN STRYLAND

For simplicity, we drop the polarization indices i, j, ... and thus ignore the tensorial properties of x(") as well as the vectorial nature of the electric fields at this time. The polarization effects will be discussed briefly in Sections 111.3 and VI.7. Assuming monochromatic fields, we take the general case involving the and cob as interaction of two distinct frequencies o,, E = - 'ea 2

E 2

iw.t + 2 eiwst

+ C.C.

whose Fourier transform is

Upon substituting this into Eq. (4) and separating only the polarization terms occurring at o,,we obtain

X

I

+ { c . c . } ~ ( o+ 0,) E,,EbEf 6(O - 0,)

Examination of Eq. (8) indicates that we can introduce an effective susceptibility Xeff defined as

+

where P = { ~ ~ ~ ~ f f E , , / 2-} 6o,,) ( o {c.c.}d(o + ma). Deriving the coefficients of nonlinear absorption and refraction from Eq. (9) is now straightforward. The complex refractive index is defined as

Assuming that the nonlinear terms in Eq. (9) are small compared with the

4 OPTICAL NONLINEARITIES IN BULKSEMICONDUCTORS

263

linear term, we can expand Eq. (10) to obtain

=no

+ i-a.C 2od

C + An + i-Aa

2wa

where no = (1 + 9?e{X(1)))1”, and it is assumed that we are operating in the transparency regime where the background linear absorption coefficient is small, a. cc Y m { x ( * ) }<< @e{X(’)}. The presence of a nonzero a. in the transparency region may be due to processes such as band tailing, lattice absorption, and free carrier absorption. We can now arrive at Eq. (1) from (i = a, b) and defining Eq. (11) by using the irradiance I , = ic&,n0(wi)(Eil2

and

The propagation through the nonlinear medium, ignoring the effect of diffraction and dispersion (i.e., pulse distortion) inside the nonlinear material, will be governed by the following equations describing the irradiance and phase of the probe beam:

and

In the literature, n2 is often used to describe the nonlinear index change due to many possible mechanisms ranging from thermal and molecular orientational to saturation of absorption and ultrafast f 3 ) nonlinearities. Here we restrict the use of n, to describe only the latter, i.e., the ultrafast boundelectronic nonlinear refraction.

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Another commonly used coefficient for describing the nonlinear index is ii2, defined as

where fi2 is usually given in Gaussian units (esu) and is related to n2 through

where the right-hand side is all in mks units (SI). However, the reader must be made aware that in the literature various symbols and definitions different from those given here may be used to describe the nonlinear refractive index. Similarly for nonlinear absorption fl is often used for cc2 when describing 2PA. The various mechanisms contributing to NLA processes, including 2PA, will be discussed in Section HI. Another important piece of information contained in this pair of equations (Eqs. 12 and 13) is that since it is known that the real and imaginary parts of the linear susceptibilities are connected through causality by Kramers-Kronig relations, we expect that there should be an analogous connection between the real and imaginary parts of the nonlinear susceptibility. We discuss these relations and the associated physics next.

6. Kramers-Kronig Relations The complex response function of any linear causal system obeys a dispersion relation that relates the real and imaginary parts via Hilbert transform pairs. In optics, these are known as Kramers-Kronig (KK) dispersion relations that relate the frequency-dependentrefractive index n(w) to an integral over all frequencies of the absorption coefficient a(w), and vice versa, that is,

where 9 denotes the Cauchy principal value. We drop the 9 notation in what follows for simplicity, although it is always implied. An interesting way of viewing these dispersion relations was given by Toll (1956), as shown in Fig. l(a). A wave train (a), consisting of a superimposition of many frequencies, arrives at an absorbing medium. If one frequency component

4

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NONLINEARITRS IN BULKSEMlCONDUCrORS

265

Transmitted Pulse

FIG. I , Pictorial representation of the need for a relation between index and loss: (a) input pulse electric field in time; (b) absorption component in time. (c) (a) - (b). After Toll (1956).

(b) is completely absorbed, we could naively expect that the output should be given by the difference between (a) and (b), as shown in (c). However, it can be seen that such an output would violate causality with an output signal occurring at times before the incident wave train arrives. In order for causality to be satisfied, the absorption of one frequency component must be accompanied by phase shifts of all the remaining components in just such a fashion that when the components are summed, zero output results for times before the arrival of the wave train. Such phase shifts result from the index of refraction and its dispersion. The KK relation is the mathematical expression of causality. We will start with a simple derivation of the linear KK relations and then proceed to obtain similar relations pertaining to nonlinear optics.

Linear Kramers-Kronig relations. In a dielectric medium, the linear optical polarization P(t) can be obtained from Eq. (3) as P(t) = Eg

slh.

R"'(t)E(t - ). d.

(19)

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MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

The response function R(')(z)is equivalent to a Green's function, since it gives the response (polarization) resulting from a delta-function input (electric field). In the Fourier domain, as given in Eq. (4), we have

where x(''(w) is the susceptibility defined in terms of the response function as

[Note that ~ ( " ( w )and R'"(7) are not an exact Fourier transform pair because of a missing 271.1 Causality states that the effect cannot precede the cause. In the preceding case this requires that E(t - z) cannot contribute to B(t) for t c (t - z). Therefore, in order to satisfy causality, R ( l ) ( r )= 0 for z < 0. An easy way to see this is to consider the response to a delta-function E(z) = E,d(z), where the polarization would follow R(')(t).This has important consequences for the relation between the susceptiblity x(')(w)and the response function R")(t), since the integration needs to be performed only for positive times. Therefore, the lower limit in the integral in Eq. (21) can in general be replaced by zero. The usual method for deriving the K K relation from this point is to consider a Cauchy integral in the complex frequency plane. However, in the Cauchy integral method, the physical principle from which dispersion relations results (namely, causality) is not obvious. The principle of causality can be restated mathematically as

i.e., the response to an impulse at t = 0 must be zero for t < 0. Here, @ ( t ) is the Heaviside step function, defined as @ ( t ) = 1 for t > 0 and @ ( t ) = 0 for t < 0. [It is also possible to use the "sign" function at this point or any other function that requires R"'(t) = 0 for t < 0.3 Upon Fourier transforming this equation, the product in the time domain becomes a convolution in frequency space:

4

OPTICAL

NONLINEARITIES IN BULK SEMICONDUCTORS

267

which is the KK relation for the linear optical susceptibility. Thus the KK relation is simply a restatement of the causality condition (Eq. 22) in the frequency domain. Taking the real part, we have

It is more standard to write the optical dispersion relations in terms of the more familiar quantities of refractive index n(o) and absorption coefficient a(w) (Price, 1964). These relations are derived in Hutchings et al. (1992) using relativistic arguments. However, if we assume dilute media with small absorption and indices, we obtain the identical result. By setting n - 1 = % ? e ( ~ ( ' ) )and / 2 a = o A n { , y ( ' ) ) / c , we obtain n(o) - 1 =-

a(0')

do'

Since E(t) and P(t) are real, n( - o)= n ( o ) and a( - w ) = a(o), which allows the integral in Eq. (25) to be written with limits from 0 to co giving the final result of Eq. (18). More rigorous derivation of Eq. (25) has been given by Toll (1956) and Nussenzweig (1972). Nonlinear Kramers-Kronig relations. Although dispersion relations for linear optics (i.e., KK relations) are well understood, confusion has existed about their application to nonlinear optics. Clearly, causality holds for nonlinear as well as linear systems. The question is: What form do the resulting dispersion relations take? The usual Kramers-Kronig relations are derived from linear dispersion theory, so it would appear impossible to apply the same logic to a nonlinear system (Hutchings et al., 1992). Since the birth of nonlinear optics, there have been numerous articles addressing the dispersion relations (Kogan, 1963; Caspers, 1964; Ridener and Goud, 1975; Bassani and Scandolo, 1991). However, the usefulness of these relations was not fully appreciated and used until recently (Sheik-Bahae et al., 1990a, 1991). The simplest way to view this process is to linearize the problem. By viewing the material plus strong perturbing light beam as a new linear system on which we apply causality; we obtain a new absorption spectrum for the material, as illustrated in Fig. 2. The linear Kramers-Kronig relation can be applied both in the presence and in the absence of a perturbation and the difference taken between the two cases. This is to say that the system remains causal under an external

268

MANSOORSHEIK-BAHAE AND ERICW.VAN STRYLAND

FIG.2. (a) Linear system, material only, probed at w' to determine the linear absorption spectrum. (b) The system is now the material plus strong light beam at o,,probed at w', to determine the changed "linear" absorption spectrum.

perturbation. We can write down a modified form of the Kramers-Kronig relation (which we also derive below specifically for an optical perturbation):

which after subtracting the linear relation between n and a leaves a relation between the changes in index and absorption:

where [ denotes the perturbation. An equivalent relation also exists whereby the change in absorption coefficient can be calculated from the change in the refractive index, but this is rarely used for the reason described below. Note that it is essential that the perturbation be independent of frequency of observation o'in the integral (i.e., the excitation [ must be held constant as o' is varied). This form of calculation of the refractive index for nonlinear optics is often more useful than the analogous linear optics relation. Absorption changes (which can be either calculated or measured) usually occur only over a limited frequency range, and thus the integral in Eq. (27) needs to be calculated only over this finite frequency range. In comparison, for the linear Kramers-Kronig calculation, absorption spectra tend to cover

4

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NONLINEARITIES fN BULK SEMICONDUCTORS

269

a very large frequency range, and it is necessary to take account of this full range in order to obtain a quantitative fit for the dispersion. Unfortunately, the converse is not true, since refractive index changes are usually quite extensive in frequency. Equation (27) has been used frequently to determine refractive changes due to “real” excitations such as thermal and free-carrier nonlinearities in semiconductors (Haug, 1988; Miller et al., 1981a; Jain and Klein, 1983). In those cases, c denotes either AT (change in temperature) or AN (change in free-carrier density), respectively. For example, this method has been used to calculate the refractive index change resulting from an excited electronhole plasma (Miller et al., 1981b) and a thermal shift of the band edge (Wherrett et d., 1988). For cases where an electron-hole plasma is injected, the subsequent change in absorption gives the plasma contribution to the refractive index. In this case, the [ parameter in Eq. (27) is taken as the change in plasma density regardless of the mechanism of generation of the plasma or the pump frequency. Van Vechten and Aspnes (1969) obtained the low-frequency limit of n2 from a similar KK transformation of the Franz-Keldysh electro-absorption effect. In this case, c is the dc electric field. Here we apply this formalism to the case where the perturbation is “virtual,” occurring at an excitation frequency wb. To the lowest order in the excitation irradiance I,, we write

This leads to dispersion relations between a2 and n,:

We note here that even when the degenerate n2 = n,(o,; 0,) is desired for a given w,, the dispersion relations require that we should know the nondegenerate absorption spectrum a,(o’; wb) at all frequencies o’. Also, the implication of Eq. (29) for the nonlinear susceptibility, on using Eqs. (12) and (13), is

270

MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

which, using the symmetry properties of x ( ~ )also , can be written as

Another way of deriving Eq. (31) in a more general form is by applying the causality condition directly to the nonlinear response R". Starting with Eq. (S), causality implies that the nonlinear susceptibility can be determined by integration over positive times only:

x exp[-i(w,z,

+ w 2 ~+2 + w,,r,)]

(32)

It is now possible to apply the method used earlier for the linear susceptibility in order to derive a dispersion relation for the nonlinear susceptibility. For example, we can write

x(")(rl,z2,. .., T")

= X(")(Z,,

r2,...)Z " ) @ ( T j )

(33)

and then calculate the Fourier transform of this equation. Herej can apply to any one of the indices 1, 2,. .., n. We also could use any number and combination of step functions; however, the simplest result is obtained by taking just one. Following the same procedure as for a linear response, we obtain

where, by equating the real and imaginary parts on both sides, we get the generalized Kramers-Kronig relation for a nondegenerate nonlinear susceptibility:

w e { ~ ( " ' ( w w2,..., w,)}

's

=-

=

O3

-m

9m{X(")(w1, w2,..., w',.. ., w")>dot w' - wj

In particular, for x(3) processes having w1 = w,, w2 = wb, and w3 = -ab, this becomes identical to Eq. (31) derived earlier. When including harmonic-

4 OPTICAL NONLINEARITIES IN BULKSEMICONDUCTORS

271

generating susceptibilies, however, Eq. (35) can be further generalized as given in Hutchins et al., (1992). The key concept described by Eq. (29) is that once we calculate the change in absorption spectrum induced by an excitation at ob, the nonlinear refraction n, can be obtained by performing the KK dispersion integration. We emphasize again that even degenerate n 2 ( o , = ob) must be derived from a nondegenerate nonlinear absorption spectrum. In the following section, the calculation of a,(o,; wb) [equivalently Aa(w,; c o b ) ] using a twoparabolic-band (TPB) model for semiconductors is described.

111. Theory of Bound-Electronic Nonlinearities Two-Band Model 1. NONDEGENERATE NONLINEARABSORPTION

In this section we calculate the nonlinear absorption originating from x(3) by including 2-photon absorption (2PA), the AC Stark effect, and Raman contributions (Sheik-Bahae et al., 1991, 1994). Widely available experimental results for degenerate 2PA serve as a calibration for the calculation. Analysis of 2PA processes requires that perturbation theory be taken to second order (Worlock, 1972). A variation of this is to use first-order perturbation theory with a “dressed’’ initial and final state where the effect of the acceleration of the electrons due to the oscillating electric field is already taken into account. We use the dipole approximation for the radiation interaction Hamiltonian:

where 2 is the vector potential, p is the momentum operator, - e is the electronic charge, and mo is the bare electron mass. We assume a two-beam interaction with both beams linearly polarized in the same direction. This last assumption allows us to write the following simple expression for 3:

-A = iilAo, cos(o,t) + ii,A,,cos(w,t)

(37)

where A,, and A,, are the vector field amplitudes at frequencies o,and o,, respectively (Note: In this section we use w , and o,instead of w, and ob.) ii, and ii, are the polarization unit vectors and are taken parallel for the calculations presented in this section. The case of orthogonal polarization will be discussed briefly in Section 111.3. Following Keldysh (1965), Jones

272

MANSOORSHEIK-BAHAE AND ERICw.VAN

STRYLAND

and Reiss (1977), and Brandi and de Araujo (1983), the initial (valence band) and the final (conduction band) states can be approximated by Valkov-type wavefunctions (Volkov, 1935):

[- ;j;

$,.,(~,3, t ) = u,,JX, 7 )exp i k -3 - -

E,,,(T) dx

]

(38)

where uCJx -7) is the usual Bloch wavefunction for the conduction (c) or valence ( 0 ) band states, and the corresponding AC (or optical) Stark-shifted energy of the states are given by

Within the effective mass approximation and using the two-parabolicband (TPB) model, the unperturbed energies of the final and ground states are given by

E,O = E ,

h2k2 +2mC

and

where k is an electron wavevector, and m, and m, are the conduction and valence band effective masses, respectively. The linear optical Stark shift of the energy bands is given by

The quadratic optical Stark effect (QSE) resulting from the coupling between conduction and valence bands due to w2 is written as AE,, = - AE,,, =

1

E;

- E:

- hw,

+ E:

where p,,. is the interband momentum matrix element:

- E:

+ hw,

4 OPTICAL NONLINEARITIES IN BULKSEMICONDUCTORS

273

In the TPB model using Kane’s k - p theory, we have that

m, = - m u x m,- EB EP

(44)

where E p = 21pJk = 0) 12/mox 21 eV is the Kane energy, which is essentially material-independent for most semiconductors (Kane, 1957, 1980). The transition rates are calculated using an S matrix formalism with (Wu and Ohmura, 1962) S =f

h

s’

dt‘ jd3it,b:(x,7, r’)Hin,t,bu(l’,7, t’)

(45)

-m

The transition rate W is then obtained from

The transition rate given by Eq. (46) is highly nonlinear and includes all N-photon transitions (N = 1, 2,. .., 00) involving one virtual interband transition followed by N - 1 self-transitions. In this analysis, we are only concerned with the x ‘ ~ ’processes. In the expansion of W therefore, we retain only the terms that are proportional to Ill? (where I j = E ~ ~ ~ C O that ~ A ~ ~ / ~ ) involve the absorption of one photon with energy hw,. This gives the change of absorption at o1due to the presence of 0,:

‘1

The physical processes emerging from this formulation are as follows. First, there is a nondegenerate 2PA that requires ho,+ ho, 2 E,. Second is an electronic Raman transition for which Iho,- hw,I 2 E, is required. In the “dressed state” formalism, these two processes are a consequence of the first-order (time varying) AC Stark shift described by Eq. (41). In addition to the absorption of one photon at wl, there is simultaneous absorption (2PA) or emission (Raman) of a photon at the excitation frequency w2. A third process is due to a combination of linear and primarily quadratic A C Stark effects. These only affect the photon absorption at w1 and hence occur for ho,2 E,. This process can be viewed as saturation of absorption due to state blocking caused by virtual carriers

274

MANSCORSHEIK-BAHAE

v v

AND ERIC

w.V A N STRYLAND

. ...... ....

............

t

if

2PA

Rcvllan

AC sterk

FIG. 3. Graphic representation of the processes involved in the nonlinear absorption in a two-band model. The small circles denote bound electrons.

generated by 0,. This effect is also referred to as virtual band blocking. All three processes described here are depicted graphically in Fig. 3. The resulting representation of a, for 2PA has the simple form

where K is a material-independent constant:

The function F, involves only the parameters x I = Ao,/E, and x 2 = Am2/ E, and reflects the band structure and intermediate states considered in the calculations. In our simple model, F , contains contributions from 2PA, Raman, and optical Stark effects. These different components are listed in Table I, and the function F , is plotted in Fig. 4 for different values of x 1 and x2.

Of the nonlinear absorption processes described here, only 2PA has been studied extensively. For over three decades, the 2PA coefficient (j? = has been measured for semiconductors and other optical solids. More recently, the magnitude, band-gap scaling, and spectral variation of 2PA in

4 OPTICAL NONLINEARITIES IN BULKSEMICONDUCTORS

275

TABLE 1 CONTRIBUTIONSTo THE NONLINEAR ABSORPTION SPECTRAL FUNCTION Fz(Xl,

Contribution

FAX 1.

Xz)

x2)

+ xz - 1)3/2

2-photon absorption XI xz > 1

+

AC Stark XI > 1

1

XI

2 ( ~, 1Hx:

-

+ x:)

+-I

8(x1 - 1)’ x:

semiconductors have been obtained using standard transmission measurements (Van Stryland et d., 1985b). The experimental data are in good agreement with the calculation presented here. We are now in a position to look at the spectral dependence of the degenerate 2PA. From Table I the 2PA contribution to the nonlinear absorption for x 1 = x2 = x is

F2=

(2x - 1)3/2 25x5

0.3 I

I

FIG. 4. The nonlinear absorption spectral function F, showing the change of absorption at w , due to three different photon energies wz.

MANSOORSHEIK-BAHAE AND ERICw.V A N STRYLAND

276

0.061

I

0.04

’

LN

0.02

0

I

I

0’

0.2

0.4

0.8

0.6

1 .o

FIG. 5. The degenerate 2PA spectral function F,(hw/E,) along with data scaled according to Eq. (51). (Data using transmittance from Van Stryland et a/., 1985b.)

The solid line in Fig. 5 shows this spectral dependence. This functional dependence, as well as the scaling, also can be derived directly from second-order perturbation theory (Wherrett, 1984). Comparison with experiment is done best by measuring the nonlinear absorption spectrum for individual materials. Unfortunately, there are few materials for which nonlinear spectra are known. One reason for this is that tunable sources with the required irradiance, pulse width, and beam quality are not typically available. Instead, we scale the material dependence using predictions of the two-band model [Eq. (48) with xI = x2 and Eq. (50)]. Now experimental data for B (B“) can be plotted by calculating the experimental value of the function F,, F;, as given below. The absolute magnitude for this function is then a fitting parameter K: F;(hofE,) = - 2 ~ 3 8 ’ K A n 0

’

Figure 5 plots scaled data for several semiconductors versus h o f E , , with the predicted dependence from the TPB model (Van Stryland et al., 1985b). Measurements of degenerate B on a number of 11-VI and 111-V semiconductors fit this model using K = 3100cm GW-’eV5” in units where E , and E, are in electronvolts (3.2 x 10-55mks) (Van Stryland et al., 1985a). This

4 OPTICAL NONLINEARITIES IN BULKSEMICONDUCTORS

1

277

10

E p( W

FIG. 6 . Log-log plot of scaled jF as a function of E,. The straight line is of Eq. (51) showing the E9-3 dependence. Adapted from Van Stryland et al. (1988).

compares favorably with the theoretical value of K = 1940cmGW-’ eV5’, (1.99 x mks) given by Eq. (49). We believe that the theory underestimates the empirical value because only one valence band (light-hole) has been considered, and the contribution of the heavy-hole band has been ignored. This figure shows 2PA turning on sharply at half the band-gap energy and then slowly decreasing for photon energies approaching the band gap. This behavior is similar to the behavior of linear absorption but shifted by a factor of 2 in wavelength. We will see later that the nonlinear refraction similarly mimics the linear refraction but shifted by a factor of 2 in wavelength. The E; dependence of 2PA is better displayed on a log-log plot scaling the data with F,, as shown in Fig. 6. This shows that even dielectric materials follow the general trend predicted by Eq. (51). We also remark that the result derived for 2PA using first-order perturbation and dressed wavefunctions is identical to that obtained from a calculation using second-order perturbation theory (Wherrett, 1984).

2. NONDEGENERATE NONLINEAR REFRACTION With the nondegenerate nonlinear absorption coefficient az(ol, oz) determined, the next step is to perform the KK transformation of Eq. (29)

278

MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

to obtain the n2 coefficient. The result of this calculation is

where the dispersion function G, is given by

and

Using the function F, given in Table I, the integration in Eq. (53) can be performed analytically. In the low-frequency limit, however, it is found that G, diverges as x, tends to zero (equivalently,o,-,0). This divergence is not unexpected, The transition rate calculation is based on A p perturbation theory, and it is well known that divergences of this order can be introduced. The equivalent E . r perturbation approach, as was shown by Aversa et al., (1994) avoids such divergences at the expense of a more intensive calculation. To examine the “infrared” divergences, the nonlinear refractive terms can be expanded in a Laurent series around w2 = 0. Because of their unphysical nature, it has been common practice to subtract the divergent terms in the series. This brute-force process of divergence removal effectively enforces a sum rule for the two-band system. The long-wavelength divergent terms for each contribution are removed separately, and the final results are set out in Table 11. The dispersion function G, is depicted in Fig. 7 as a function of x 1 = h o , / E , for various excitation photon energies x2 = h o , / E , . By examining the terms in Table I1 individually, we can determine their relative contribution to n2 in different spectral regimes. A general trend is evident in all the curves: n2 is nondispersive in the infrared regime (hw, << Eg), where the 2PA and Raman terms contribute ajmost equally. It then reaches a two-photon resonance at hw, + hw, z E,, where the peak positive value is attained. Above the two-photon resonance, n2 becomes “anomalously” dispersive and ultimately turns negative. The negative behavior of n, is caused primarily by the 2PA and quadratic AC Stark effect components as h a , approaches E,. Recent measurements of nondegenerate n2 with the semiconductors ZnS and ZnSe, as shown in Section VI.7, show excellent agreement with this predicted behavior (Sheik-Bahae et al., 1994, 1990a).

4 OPTICAL NONLINEARITIES IN BULKSEMICONDUCTORS

279

TABLE I1 THENONDEGENERATE DISPERSION FUNCTION FOR THE ELECTRONIC KERR COMPONENT OF n2, CALCULATED BY A KK TRANSFORMATION OF F , (Table I)

where H(x,, x ) -

1 -

- 26XfX24

5

16 -x:x:

9 + -x:x: 8

9

- -x2x: 4

3

- -x; 4 -

1

- x1)-3/2

>

AC Stark 1 4 +--4 -_-2 x1 f x 2

x:

x:

1 - x:) + 2~:(3~: x:(x: - x:)z 29x:x:

- 2~:(3~: - x:) x:(x:

- xi)’

x: [(l

x,

[(l - x

1

- x1)-1/2 - (1 + x1)-1/21 x: - x: p

+ (1 + x2)’/2] + + x1)’/2]

[(l - x1)1/2 (1

Note: The infrared divergent terms associated with each contribution have been removed. The last term is due to the degenerate AC Stark effect, which is the limit of the nondegenerate GTS‘Prkas xt + x2.

While knowledge of the dispersion of the nondegenerate n, may be useful for applications that employ harmonic generation crystals, for example, the major practical interest concerns the degenerate n2(w,= 0,).In the remainder of this chapter we focus our attention on this quantity n2(o). The dispersion and band-gap scaling of n 2 ( o ) are given by Eq. (52) with

280

MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND 0.2 r

0.1

t I

\

0.0 - . . ..... .. .

-0.1

A

x2=0.2 x2=0.8

x,=O.S

.... .. ..... ... .. .

,

-

-0.2 0.0

1

0.2

.

1

0.4

.

1

0.6

.

1

0.8

.

1.o

FIG. 7. The nondegenerate dispersion function G,(x,, x2) as a function of probe photon energy ( x , ), calculated for three excitation photon energies.

w 1 = 0,= w and nol = nol = no. As was the case with the nondegenerate

n2, the two features of interest in the nz coefficientare the band-gap energy scaling and dispersion. The former is characterized by an EB4 scaling, while the latter exhibits a sign reversal as hw approaches E, and also a twophoton resonance enhancement. We can now use the results of the KK integral for G2(w1; 0,)to give the degenerate G, and compare with experimental data. As for /?, measuring the frequency dependence of n2 for a single material is very difficult, and few data exist. Scaling data according to Eq. (52) (degenerate x 1 = x2 = x) allows us to compare experimental measurements made on different materials. Figure 8 shows the dispersion (GJ of n2 as predicted by Eq. (52) along with data (n5) from several experiments scaled according to Sheik-Bahae et al., (1991)

It is worth mentioning that the band-gap scaling predicted by the TPB model also can be obtained using a quasidimensional analysis. Such an analysis was used by Wherrett to obtain the band-gap energy scaling of B. This scaling directly gives the Ei4 scaling of n,, since n2 is directly proportional to xt3), and Wherrett (1984) showed that f 3 ) aE i 4 .

4 OPTICAL NONLINEARITIES I N BULKSEMICONDUCTORS

a

-a

0.05

N

0.00

wY

r“ 2

281

-0.05 -

-0.1

g.O

-0.20

I

,

0.2

,

!

0.4

,

0.6

,

I

,

1.0

0.8

1

t -1’08A0 -0.80

0.85

A

0.90

0.95

,!A0

(b) A 6 FIG. 8. A plot of experimental values of the nonlinear refractive index n; scaled according to Eq. (55) versus x = ho/E,,. The solid line is the two-parabolic-band-modelprediction for the dispersion function G,. (a) Circles from Adair (1989), diamond from Ross et al. (1990), and squares from Sheik-Bahae et al. (1991). (b) An extension of (a) for frequencies near the band edge (expanded scale); triangles from LaGasse et al. (1990). Adapted from Sheik-Bahae et al. (1991).

The hidden Ei4 scaling can be displayed more conveniently on a log-log plot of n, scaled by the dispersion function G,, as in Fig. 9. Here it is seen that the nonlinear index varies from 7.6 x 10-”cm2/W for MgF, at 1.06pm to -3.3 x 10-’2cmz/W for AlGaAs at 810nm and 2.8 x 10-’3cmZ/W for Ge at 10.6pm. Note, for example, that although the

282

MANSOORSHEIK-BAHAE AND ERICw.V A N

lo-"

I ' ' 1 ' 1

10

1

STRYLAND

I

Eg iev)

FIG. 9. Log-log plot of scaled n; as a function of E, showing the El4 dependence (straight line). The AlGaAs data are taken from LaGasse et al. (1990), and much of the large-band-gap data are from Adair et a/. (1989). Adapted from Sheik-Bahae et al. (1991).

measured values of n2 for ZnSe at 1.06 and 0.532pm have different signs, both measurements are consistent with the scaling law. The prefactor K', as given by Eq. (54), is 3.2 x lo-" (mks units). If we use the K as determined by a best fit to the 2PA experimental data, we find K' zz 5 x lo-*' (mks). This compares well with K' ;2: 9 x lO-"(mks) determined by an overall best fit to the experimental values of n, in semiconductors, as shown in Fig. 9. As in the 2PA case, it is more convenient to define the K ' factor in mixed units such that n, is given by square centimeters per watt and E, and E, are given in electronvolts. In this case, K' z 6 x lo-" ( c ~ ~ / W ) ( ~ Vcorresponds )"~ to the overall best fit to semiconductor n2 data. As pointed out for the 2PA case, the factor of 2 to 3 discrepancy between theory and experiments can be attributed in part to using a single valence band (light hole) and ignoring the transition originating from the heavy-hole valence band. Moreover, as described by SheikBahae el a/. (1994), inclusion of electron-hole Coulomb interaction by multiplying the F , function by a generalized exciton enhancement factor will further improve the agreement between theory and experiment. A number of theoretical efforts to extend the simple TPB model to a more compex system have been reported. For example, Hutchings and Wherrett (1994, 1995) used a Kane four-band structure to calculate the dispersion of

4

OPTICAL NONLINEARITIES IN

BULKSEMrCONDUCTORS

283

n, in zinc blende semiconductors. Aversa et al. (1994), as mentioned earlier, used E .r instead of A * p perturbation to calculate n2 in order to avoid the unphysical infrared divergences. Interestingly, the resulting dispersion, magnitude, and scaling properties of the preceding theories are nearly identical to the simple TPB theory presented here. Efforts also have been made to extend the n, theory to include quantumconfined structures. Khurgin (1994) calculated n2 for quantum well and quantum wire structures by applying the KK transformation to the calculated two-photon absorption spectrum. Cotter et al. (1992) also calculated n, in quantum-confined semiconductor nanocrystals. They obtained a dispersion function that is similar to bulk materials but changes sign (i.e., becomes negative) at longer wavelengths due to an enhanced quadratic Stark effect.

AND ANISOTROPY OF X‘3’ 3. POLARIZATION DEPENDENCE

The anisotropy of the band structure, reflecting the crystalline structure of a particular semiconductor, manifests itself in the NLO properties by making n, and a, (or /?) anisotropic as well. A number of theoretical considerations as well as experimental measurements have been reported dealing primarily with the anisotropy of two-photon absorption in semiconductors belonging to various symmetry groups (Rader and Gold, 1968; Dvorak et al., 1994; De Salvo et al., 1993; Balterameyunas et al., 1982). Even in an isotropic material, such as polycrystalline semiconductors, the effect of the polarization of the incident electric field can still be studied by conducting induced-anisotropy experiments. Such experiments include polarization-dependent four-wave mixing, excite probe, and measurements of linear/circular dichroism. A simple extension of the TPB model of Section 111, including the effect of the heavy-hole (hh) valence band, can provide the necessary insight into the observed polarization effects (Sheik-Bahae et al., 1995). The basic principle behind this theory lies in the k-space orientation of the momentum matrix element pcowith respect to the lattice wave vectorx. Within Kane’s k - p formalism (Kane, 1980), xlljcofor the light-hole-toconduction-band (lh-c) transitions (as used in Section III.l), whilex Ijcofor the hh-c transitions. The latter orientation can be interpreted as the reason for the bare electron effective mass (mo)of the hh band because k - p coupling between that band and the conduction band vanishes. The significance of this k-space symmetry becomes apparent if we examine the expression for the transition rate due to the Raman effect and 2PA as obtained from Eq. (46):

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MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

(56) where unit vectors Z, and Z, represent the polarization of the two optical fields at frequency w l and w2, respectively. The sequence of transitions in Eq. (56) is typical of a two-band case in which a photon is absorbed in an interband transition followed by an absorption (+ sign) or emission ( - sign) of the second photon in an intraband process (self-transition). Using the k-space orientation properties o f x and pCu,as discussed earlier, Eq. (56) leads to distinctly different polarization dependences for the two band pairs. For example, for the hh-c system, the following relationship for the degenerate 9rn{~‘~’) is derived:

whereas the Ih-c system follows the x1122= x1212= xlZzl= x l l 1 1 /sym3 metry. In contrast, the nondegenerate transition rate due to the QSE involves only interband transitions:

WQSEa

1

\(a’* *PCU)ca2.PC”),,)l2S(hO 1 - E,,)d~

(58)

The lack of self-transitions makes the symmetry relations for xQSEthe same (xllt2 = x~~~~ = x~~~~ = x1,11/3) for both band pairs. In adding up all the contributions of the two-band pairs for both refractive and absorptive processes, one obtains the dispersion of the x ‘ ~ ’tensor for an isotropic three-band system. The result of this calculation can explain the observed polarization dependence in four-wave mixing and Z-scan experiments (Sheik-Bahae ef al., 1995). Figure 10 shows the calculated dispersion of circularflineardichroism [defined as the ratio of n2 (circular pol.) to n2 (linear pol.)] as compared with experimental results obtained with some semiconductors as well as dielectrics. Similar dispersion of the dichroism also was derived by Hutchings and Wherrett (1994) using a Kane four-band model. IV. Bound-Electronic Optical Nonlinearities in Active Semiconductors The ultrafast NLA and NLR in semiconductor laser amplifier (SLA) waveguides have been the subject of recent studies using femtosecond self-phase modulation and pump-probe techniques (see Volume 59, Chap. 2)(Hultgren and Ippen, 1991; Hultgren et al., 1992; Hall et a!., 1993; Fisher

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285

FIG. 10. The dispersion of the polarization dichroism of n, measured using Z-scan.The solid line is from the three-band theory, and the dashed line represents the two-band model. The divergence near h o / E z 0.65 corresponds to the zero crossing of n2 (linear polarization) at that wavelength. Figure from Sheik-Bahae et a/. (1991).

et a/., 1993; Hong et al., 1994; Grant and Sibbet, 1991). These delicate measurements revealed some intriguing features, including large, near-instantaneous, bound-electronic nonlinear refraction as well as transient carrier effects. The fast, bound-electronic nonlinearity, coupled with the low loss associated with the SLA, makes this device very attractive for possible use in all-optical interchange applications. The SLA exploits ultrafast nonlinear refraction without suffering concomitant absorption that plagues the approaches based on passive materials. Since in this chapter our interest lies in the ultrafast n2, we are not immediately concerned with long-lived processes that result from net carier excitation or deexcitation. As discussed in Volume 59, Chapter 2, in practice, operating at the transparency point of the device, adjusting the wavelength, and/or controlling the injected current density of the SLA can turn off such processes. However, even at the transparency point ( h o = Eg), there exist real excitation processes that accompany the bound-electronic component. These processes include spectral hole burning (SH B) and free-carrier absorption, both of which lead to carrier heating that shifts the transparency point. Here we briefly discuss the results of the theory of bound-electronic n2 in active semiconductors (Sheik-Bahae and Van Stryland, 1994). This formal-

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-0.0

CY -0.2

0.9

1

.c

1.1

1.2

hdEg

FIG. 1 I . Calculated dispersion function G, for various injected current density levels. The dashed line is for passive material. The transparency point of each curve is marked with a vertical bar. Figure from Sheik-Bahae and Van Stryland (1994).

ism is a simple extension of the TPB theory describing bound-electronic processes in passive semiconductors presented in Section 111. The nonlinear refractive index n, in the SLA is described by an equation similar to Eq. (52), except that the dispersion function depends not only on hw/E, but also on the lattice temperature (kT/E,), quasi-Fermi levels set by the injected current density, and broadening due to polarization dephasing (h/T,E,). Figure 11 depicts the calculated dispersion function G, for various current injection levels plotted as a function of photon energy normalized to the passive band gap. The dynamic transparency point changes with current density, as indicated by vertical bars in the figure. The sign and magnitude of the predicted n, values are in close agreement with existing experimental results (Haltgren and Ippen, 1991; Haltgren et al., 1992; Grant and Sibbet, 1991). A straightforward comparison can be made between the calculated spectral dispersion function G , for a SLA at the transparency point and the value of G, for passive material operating just below half the band gap. The comparison indicates that more than an order-of-magnitude enhancement of this function can be obtained with active materials. Note, however, that the nonlinear refraction is determined by the product of the factors G, and E i 4 . For a given wavelength, SLA operation requires a band gap that is smaller by about a factor of 2. Therefore, for a fixed wavelength, the two contributions will enhance n, for an SLA by more than 2 orders of magnitude compared with a half-band-gap

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passive switch (see Section VII.l on all-optical switching). Recent optical switching experiments using SLA devices have indicated nearly an order of magnitude lower switching power compared with passive NLDC devices (Lee et al., 1994). Further optimization and systematic study of SLA nonlinearities are expected to lower the switching threshold further. Parasitic effects arising from the dynamics discussed earlier introduce complications that need to be considered. A comprehensive theory that selfconsistently unifies bound-electronic effects with spectral hole burning and carrier heating is presently not at hand.

V. Free-Carrier Nonlinearities Besides ultrafast bound-electronic nonlinear effects discussed so far, additional NLA and NLR effects can arise from free carriers generated by multiphoton absorption in the transparency region. These nonlinearities are distinguishable from f 3 ) effects because they are cumulative (with a decay given by the carrier lifetime) and appear as higher-order processes (if the carriers are produced by linear absorption, the resulting nonlinear response is third order, see Chap. 1 in this volume and Chap. 5 in Volume 59). In this chapter we discuss the free-carrier effects mainly to point out their role in complicating the f 3 ) measurement process. In fact, the discrepancies of measured values of 2PA coefficients reported in the literature often can be understood by understanding these effects (Van Stryland and Chase, 1994). For example, in a single-beam experiment, if significant carrier densities are created, the nonlinear absorption equation [Eq. (14) for a single beam] must be modified as follows: dl

- = -aI - PI2 - @,ANe dz

+ ohAN,)I

(59)

where B , , ~is the free-carrier absorption (FCA) cross section (units of square centimeters) for electrons and holes, respectively. In many semiconductors described by the Kane model, free-hole absorption dominates (i.e., o h >> 0,) due to strong inter-valence band absorption. In a two-band approximation, we take ANe = A N h = A N and define CJ = B, + oh.Assuming 2PA is the only mechanism for generating carriers, and neglecting population decay within the pulse and spatial diffusion, the carrier generation rate is given by dAN --PI2 --

dt

2ho

The combination of Eqs. (59) and (60) shows a fifth-order nonlinear response of the loss for carriers generated by 2PA (i.e., N is proportional to

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MANWORSHEIK-BAHAE AND ERICW.VAN STRYLAND

12, leading to an Z3 dependence of the loss). The fifth-order response comes from the combination of Y r n ( ~ (followed ~)) by a ~ ( lprocess, ) either absorption (Yrn{x")))or refraction (We($'). Equation (60)is only valid for pulses short enough that carrier recombination, decay, and diffusion can be ignored. This shows one of the simplifications afforded by using short optical pulses for determining /3 (or n2). Another advantage is that short pulses minimize the effects of FCA (and the associated free-carrier refraction), since the energy for a fixed irradiance is reduced [and the less energy, the fewer carriers created, as seen by the temporal integral of Eq. (60)].The FCA term in Eq. (59) can range from negligible to dominant depending on the semiconductor, wavelength, irradiance, and temporal pulsewidth. For example, for InSb, the FCA terms in Eq. (59) actually dominate the overall loss even for 100-ps, 10.6-,urn pulses (Hasselbeck et al., 1997). Knowledge of the free-carrier absorption coefficient 0 allows relatively simple modeling of the overall loss; however, 0 often must be determined empirically. Even in situations where the free-carrier losses can be made negligible (e.g., for short pulses), index changes due to the carrier excitation (so-called free-carrier refraction, FCR) can still be significant. This FCR is not simply calculated via KK from the added FCA spectrum. This turns out to be a small contribution to the total NLR. The dominant NLR is instead calculated from the saturation of the interband linear absorption spectrum resulting from the redistribution of electron population. A similar process occurs in a laser where th6 index change due to gain saturation leads to frequency pulling of the cavity modes (Meystre and Sargent, 1991). The method of carrier excitation is irrelevant to the resulting index change. The removal of electrons from the valence band (creation of electrons in the conduction band) reduces the linear absorption for wavelengths near the band edge (band blocking). This is referred to as the dynamic BurnsteinMoss shift (Moss, 1980; Burstein, 1954). Carrier-carrier scattering tends to thermalize the carrier distribution on the time scale of -Ips, while recombination times are much longer. Therefore, there is a quasi-equilibrium distribution of carriers that reduces the linear absorption by removing potential interband transitions and, via causality, changes the index. The NLR is calculated from the changed absorption spectrum according to Eq. (27). The free-carrier NLR has a negative sign in the transparency region of semiconductors (from the reduced absorption), leading to beam defocusing. In experiments where the carriers are created by linear absorption with near-gap excitation, this NLR can be huge (Miller and Duncan, 1987). Here, where the excitation is by 2PA, we are well below the frequency where the index changes are large, but the effects can still be comparable or even larger than the effects from n,. The importance of understanding the free-carrier nonlinearities in the transparency region is twofold. On the one hand, it may be usefu) for applications such as optical limiting (see Section VIT.2). On the

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other hand, it can complicate the measurement or mask the usefulness of the bound-electronic effect in ultrafast applications such as optical switching. The effect of these free oscillators on the phase is proportional to the density of created carriers:

dCg - kn21 + ka,AN

_.-

dz

which includes the effects of the bound-electronic n, as well as the freecarrier refractive coefficient r ~ , (units of cubic centimeters). The k in the second term is sometimes dropped to give the refractive cross section in units of square centimeters. This fifth-order nonlinear refraction can be seen in measurements of the induced phase distortion, as shown in Fig. 12. This figure shows the index change divided by the input irradiance I, as a function of I, in ZnSe at 532nm, where it exhibits 2PA. The index change is calculated from the measured phase distortion introduced on the beam through propagation in bulk samples. For a purely third-order response, An = n21,, this figure would show a horizontal line. The slope of the line in Fig. 12 shows a fifth-order response, whereas the intercept gives n2 (note here that it is negative). The interpretation of this fifth-order response as defocusing from carriers generated by 2PA is consistent with a number of experimental measurements including degenerate four-wave mixing measurements (Canto-Said el al., 1991) and 2-scan and time-resolved two-color Z-scan measurements, as discussed in Section VI (Sheik-Bahae et al., 1992; Wang et al., 1994). 25 r

0.0

7

0.5

1.0

1.5

2.0

2.5

I ( GW/cm2) FIG.12. A plot of the ratio of the change in refractive index to irradiance as a function of irradiance for 532-nm picosecond pulses in ZnSe. Figure from Said et a/. (1992).

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MANWORSHEIK-BAHAE AND ERICW. V A N STRYLAND

We next look briefly at two different band-filling (BF) models describing this nonlinear refraction. The first model (BF1) is attributed to Aronov et al. (1968) and Auston et al. (1978), and the second is the dynamic Moss-Burstein model with Boltzmann statistics (BF2) (Moss, 1980; Miller et al., 1981b; Wherrett et al., 1988). In these theories, the change in refraction due to carriers is independent of the means of carrier generation (see Chap. 5 in Volume 59 for more information on carrier nonlinearities). In the BF1 model, the nonlinear refraction due to free carriers is calculated directly from the real part of the complex dielectric function. The creation of a density AN of free electrons in the conduction band is accompanied by an elimination of a density N of bound electrons in the valence band. The former is often referred to as the Drude contribution, whereas the latter is referred to as a Lorentz contribution to the change in the dielectric constant. The overall change in the index of refraction is given by Auston et al. (1978) An(w; AN) = a,AN = -

ANe’ E,2 2Eonow2m,, E,’ - ( h ~ ) ~

where m,, is the reduced effective mass of the electrons in the conduction band and the holes in the valence band. In the BF2 model, as was originally introduced by Miller et al. (1981), the free carriers block the absorption at frequencies higher than the energy gap by filling the available states in the conduction and valence bands. This model uses a Kramers-Kronig integral on this change in absorption. The total change in the index of refraction using a three-band model, including contributions from electrons, heavy holes, and light holes, is given by (Wherrett ef al., 1988) as

where

Jii =

x’ exp( - x’)

E - ham,, a , .= 9” kBT mi

dx

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where mo is the free electron mass, k, is the Boltzmann constant, T is the temperature in degrees Kelvin, and E , = 21p,,(k = O))2/mois the Kane energy, as discussed in Section 111.1, and is approximately 21 eV for most semiconductors (Kane, 1980) AN and A P represent the photogenerated electron and hole densities, and the subscripts c, h, and I represent the conduction, heavy-hole, and light-hole bands, respectively. Similarly, mi represents the effective mass of the band j, and mii denotes the reduced effective mass of the ij band pair. The dummy subscripts i and j represent c, h, or 1. APh and AP, are given by (Wherrett et al., 1988)

Expression (63) (with Eq. 65) is an approximation adequate for nearresonance radiation. Off resonance, as in 2PA, we find that J , should be replaced by F,, where F is defined as

- ho + J (

mCiE

F i j = - 2 J ( -~ ~

T

)

m,, E, + hw ~ ~ k,T ) +

)J

(68) ( ~

For hw z EB and E, >> kBT, the first and third terms in Eq. (68) are very small compared with the second term; thus it is reasonable to neglect them (Miller et al., 1981; Wherrett et a/., 1988). In 2PA experiments, E, L ho is comparable with E,, and all three terms in Eq. (68) need to be retained. The electron's contribution to the index change is the first term in Eq. (63) (AN,), and this includes blocking caused by electron transitions from the heavy-hole band and the light-hole band in addition to the change in the electron population in the conduction band. The other two terms give the contributions of the holes. Calculations of the free-carrier refraction using both models give good agreement with data taken on the semiconductors GaAs and CdTe at 1.06pm, and ZnSe at 532 nm using picosecond pulses (see Fig. 12) (Said et al., 1992). For these materials, both models work well, since the change in the index of refraction from transitions between the light-hole band and the conduction band (electron blocking, light-hole blocking, and free light-hole generation) contributes only about 30% for these semiconductors. Thus it is reasonable to use the approximation of a two-band model where only transitions from heavy-hole band to conduction band are considered. For these materials, the low-temperature condition or Ihw - E,I >> k,T is satisfied; for example, in the worst case of GaAs, Jhw- E,I = 0.25 eV, and at room temperature, k,T z 0.025 eV. Examining J , in Eq. (65), aij >> 1, yielding .Iij z x"2/4a. Substituting this value for J ,

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MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

in Eq. (68), F , is proportional to x2/(1 - x’), where x = ha/E,. Assuming a two-band model and substituting F , for J , in Eq. (65) shows that the change in the index of refraction due to the carrier transition blocking is

An cc

1 E,” - (ha)’

having the same frequency dependence as the enhancement factor in the BF1 model. This is expected because the same physical mechanism is used in both calculations. Following the scaling rules applied for describing n, and /3 in Section 111, it is helpful to write a similar relation for free-carrier refraction, namely, a,. For example, by replacing the effective mass parameter by moEg/E,, Eq. (62) can be reexpressed as (Wang et d.,1994)

where A = h2e2/2com,= 3.4 x 10-”cm3eV2, and H(x) = [x’(x’ - 1 ) I - l is the free-carrier dispersion function. Figure 13 compares this dispersion function with some experimental data properly scaled using A “5 2.3 x 10-22cm3eV2.The experimental procedure used for these measurements is given in Section VI.7. Another theory of free-carrier nonlinearities as given by Banyai and Koch (1986) includes the effects of electron-hole Coulomb interaction, plasma screening, and band filling. This theory has been shown to have good

I

FIG. 13. A plot of H ( x ) versus x showing free-carrier refraction for three semiconductors as compared with theory (solid line). Figure from Wang el al. (1994).

4 O ~ C ANONLINF~ARITIES L IN BULKSEMICONDUCTORS

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agreement for near-gap excitation. A quantitative comparison of the predictions of this theory with data taken at frequencies where the excitation is well below the band edge shows poorer agreement (Said ef al., 1992).

VI. Experimental Methods There are a number of experimental difficulties that need to be addressed when attempting to determine the value of ultrafast nonlinearities, fl = or n2. For example, an examination of the literature on reported values of j?for the single semiconductor GaAs shows well over a 2 order of magnitude variation in the reported value over the past four decades. As mentioned in the preceding section, competing effects of free carriers could easily lead to an overestimation of fl as well as an incorrect value of n2. Using shorter optical pulses minimizes these and other possible cumulative effects; however, even if the cumulative effects are negligible, nonlinear refraction from n2 can still affect measurements of fl, as can fl in measurements of n2. In addition, laser output pulses having unknown temporal or spatial modulation can lead to an underestimation of the irradiance. Therefore, careful characterization of the laser output is necessary. In “thick” samples, beam propagation can lead to irradiance changes from induced phase shifts within the sample. This can be quite difficult to model and properly taken into account. It is normally advisable to work in the “external self-action” (Kaplan, 1969) regime or thin-sample limit so that beam propagation effects within the sample can be ignored. This greatly simplifies interpretation of data because the equation describing nonlinear absorption can be separated from that describing nonlinear refraction, as has been assumed previously in this chapter; i.e., these equations become Eqs. (14) and (15) [or Eqs. (59) and (61) if carrier nonlinearities are included]. Note that since the nonlinear phase shift depends on the irradiance, Eq. (14) (or 59) must first be solved for I(z) in order to solve Eq. (15) (or 61). Even if the sample satisfies the thin-sample approximation, nonlinear refraction has been known to refract light so strongly after the sample that the detector may not collect all the transmitted energy. This again leads to an overestimation of the nonlinear loss. All the preceding effects can contribute to erroneous values of the nonlinear coefficients. Several experimental techniques are available for measuring the boundelectronic nonlinear response of semiconductors, i.e., fl and n2. We will only briefly discuss a few such methods: transmittance, beam distortion, degenerate four-wave mixing (DFWM), pump/probe techniques, interferometry, and Z-scan along with its derivatives. In general, it is difficult, if not impossible, with any single technique to unambiguously separate the differ-

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MANSOORSHEIK-BAHAE AND

ERIC

w.V A N STRYLAND

ent nonlinear responses. These techniques are sensitive to several different nonlinearities at once. Usually several different experiments are necessary, varying parameters such as irradiance and pulse width, to unravel the underlying physics. Clearly, reducing the pulse width, for ultrafast nonlinearities should result in a measurement of the same value of /? and n, for the same irradiance, whereas slower nonlinear responses will change as the pulse width approaches the response time. Unfortunately, from the standpoint of characterization, ultrafast and cumulative nonlinearities often occur together in semiconductors, so a simple separation is not possible 1. TRANSMIITANCE

Single-beam direct transmittance measurements have been a primary method for determining fi in semiconductors (Van Stryland et al., 1985a; Bechtel and Smith, 1976). Plots of the inverse transmittance versus irradiance are nearly straight lines with the intercept determined by a. and slope proportional to 8. This is seen by solving Eq. (59) neglecting carrier losses, i.e., a = 0, giving

Integrals over the spatial and temporal beam profiles tend to slightly reduce the slope of these plots, as shown in Fig. 14. This figure shows the inverse transmittance of collimated 532-nm pulses incident on a 2.7-mmthick sample of chemical vapor deposition grown ZnSe as a function of peak on-axis irradiance (Van Stryland et al., 1985a, 1985b). Great care must be taken to ensure that all transmitted light is collected. Two curves are shown for pulse widths of 40 and 120ps (FWHM). The fact that these two curves lie on top of one another indicates that the cumulative effects of free-carrier absorption are negligible for these pulse widths, and a value of /? can be reasonably deduced from these data as shown in Figs. 5 and 6. Longer pulse widths show a clear deviation due to FCA.

2.

BEAMDISTORTION

Measurements of n, also can be performed in transmission by monitoring the beam distortion that occurs on propagation (Williams et al., 1984). Figure 15 shows the beam distortion in the near field introduced in ZnSe by picosecond 532-nm pulses by the combined effects of 2PA, boundelectronic n2, and free-carrier refraction (FCR). As determined from a series

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. ZnSe

3.2

-

0.8

-

z

A

12ops

I

1.2

-

Jj0.8

-

e E

3

2

W

0

o 0.4

4

G: 0.0 -0.4

'-

I -2.4

I

I

I

-1.2

0.0

1.2

2.4

Radius (mm) FIG. 15. A one-dimensional spatial profile in the near field of a picosecond 532-nm pulse transmitted through ZnSe. Figure from Van Stryland (1996).

2%

MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

of other experiments (Z-scan and DFWM), both n2 and the FCR lead to self-defocusing and contribute about equally to the self-lensing shown in Fig. 15 (the solid line is the theory using parameters deduced from other experiments). The sensitivity of this experiment is limited. For example, peak on-axis optical path length changes need to be greater than approximately ,4/4 in order to see changes to a Gaussian beam when propagated to the far field. It is also difficult to separate these different contributions with only beam distortion measurements. Even 2PA alone leads to beam shape changes with propagation. For example, a Gaussian beam is spatially broadened after propagation through a 2PA material because the center portion of the beam is preferentially absorbed, and therefore, the diffraction is reduced. This effect could be mistaken for self-focusing.Sheik-Bahaeer al. (1990) and Hermann and Wilson (1993) give details of the modeling of propagation for samples that satisfy the external self-action criteria. Whereas Chapple et al. (1994), Sheik-Bahae et al. (1990), Hermann and McDuff (1993), and Tian et al., (1995), give information on modeling methods for thick samples

3. EXCITE-PROBE Pump-probe (or excitation-probe) measurements are useful for studying the temporal dynamics of nonlinear absorption (Shank er al., 1978). In these experiments, an excitation pulse (pump) excites the sample (changes its optical properties), and a probe pulse, spatially coincident with the pump, detects the changes in the optical properties as a function of time delay after the pump. The change in transmittance of a weak probe pulse as a function of time delay after excitation by a short optical pulse allows slow and “fast” nonlinear responses to be separated. The probe is usually derived from the excitation pulse. In the case of equal frequencies (degenerate), the probe is a time-delayed replica and must be separated from the pump either spatially or by using a different polarization. Nondegenerate 2PA can be determined by frequency shifting either the pump or probe, allowing spectral separation of the probe transmittance. Determining whether or not the “fast” response is due to 2PA depends on the laser pulse width used. For picosecond/ subpicosecondexcitation, one can be reasonably certain that a signal is from multiphoton absorption if it follows the input pulse in time and shows an increasing loss. If in addition the loss vanes according to Eq. (70) (a third-order nonlinearity), it is likely due to 2PA. Nondegenerate nonlinear absorption spectra also have been measured, often using a fixed-frequency laser pump combined with a white-light probe such as the output of a flashlamp(Hopfield et al., 1963). In such a case, the

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297

temporal extent of the white-light source is usually much longer than the laser pulse and is often measuring the spectrum of cumulative nonlinearities, which can be different from the initial 2PA spectrum. The advent of femtosecond white-light continuum generation has allowed nondegenerate spectra to be taken on short time scales, where the ultrafast response dominates (see Section VI.8) (Bolger et al., 1993). Again, interpretation of the nonlinear response is made difficult by the fact that these pump/probe methods are sensitive to any induced change in loss; however, most induced phase shifts will not give rise to a measurable signal. An experimental geometry that allows index changes to generate large signals is the optical Kerr-Gate. This is a form of pump/probe experiment where induced anisotropy leads to polarization changes (Maker et al., 1964). As discussed next, three-beam interactions can produce a fourth beam through NLA and/or NLR. 4. FOUR-WAVE MIXING

Four-wave mixing, where three beams are input to a material and a fourth wave (beam) is generated, can be used for determining the magnitude of a material’s nonlinear response and its response time. If the response is known to be third-order and ultrafast, 1 f 3 ) 1 can be determined along with some of its symmetry properties by varying the relative polarizations of the input beams (as well as by monitoring the polarization of the fourth wave). In addition, the frequencies of the input beams can be changed independently to determine the frequency dependence of the nonlinear response, but this can result in the need for a complex geometry to satisfy phase-matching requirements. Equal frequencies are often used, resulting in a much simpler geometry for phase matching, and this is referred to as degeneratefour-wave mixing (DFWM). Figure 16 shows one simple geometry for DFWM where two of the input beams (the forward and backward pumps) are oppositely directed. If these beams are nearly plane waves (i.e., well collimated), this geometry ensures phase matching for any third input beam (the signal). Introducing delay arms into each of the beams also allows the temporal dynamics of the nonlinearities to be measured for short optical input pulses. A particularly useful measurement (see Fig. 16) is to monitor the energy of the fourth beam (so-called phase-conjugate beam) as a function of the time delay of the perpendicularly polarized backward pump (signal and forward pump have the same linear polarization) (Fisher, 1983). Figure 17 shows the results of this experiment performed on a sample of ZnSe using 30-ps, 0.532-pm pulses (Canto-Said et al., 1991). Clearly, two very distinct nonlinearities are evident from this figure. Near zero delay, a

-

298

MANSOORSHEIK-BAHAE AND ERICw.VAN STRYLAND BS2

0S3

NdYAG Laser

FIG. 16. DFWM geometry to allow temporal dynamics measurements. Detector D, monitors the conjugate beam energy. Figure from Canto-Said et al. (1991).

large, rapidly decaying signal is seen that follows the input pulse. At longer delays, we observe a more slowly decaying signal. To better understand the two nonlinear regimes, irradiance-dependent experiments can be performed. The inset in Fig. 17 shows a log-log plot of the DFWM signal versus the total input irradiance (all three input beams are varied simultaneously) at two different delay times. The zero-delay curve gives a power dependence of 3.1 +_ 0.2, indicative of a third-order nonlinearity. The curve for a delay of 200ps shows a power law dependence of 5.0 k 0.2. This is the fifth-order carrier nonlinear refraction discussed in Section V. Here, a modulated carrier density, created by 2PA from the interference of the copolarized forward pump and signal beam, creates a modulation of the refractive index (FCR) that scatters the backward pump into the fourth beam. In principle, free-carrier absorption also will contribute, but other experiments (see Section VI.7) have shown that FCR dominates for these pulse widths. For longer pulses, free-carrier absorption also would contribute to the fifth-order signal. The carrier grating then decays due primarily to diffusion of the carriers between interference fringes as well as some decay by recombination. Studies in CdTe at 1.06pm, where this material exhibits 2PA, reveal the same basic behavior (Canto-Said et al., 1991). One of the difficulties in the interpretation of DFWM data for third-order nonlinearities is that the signal is proportional to Ix(3)12 = i P e ( ~ ' ~+' )4 m { ~ ' " ' j 1 ~and , so 2PA and n, both contribute. Separating the effects is difficult without performing additional experiments. Also, as seen in Fig. 17, higher-order nonlinearities also can contribute, making separation of absorptive and refractive effects difficult. +

299

4 OPTICAL NONLINEARITIES IN BULK SEMICONDUCTORS 24

20

16

4

I

0

-100

I

I

I

0

100

200

I

1

300

Backward Pump Delay (PSI FIG. 17. DFWM signal in ZnSe for temporally coincident, copolarized,forward pump, and probe as a function of the time delay of the perpendicularly polarized backward pump. The inset shows a log-log plot of the output signal as a function of the input (all three inputs varied) for (a) zero temporal delay and (b) -200-ps delay. Figure from Canto-Said et al. (1991).

5.

INTERFEROMETRY

A number of interferometric methods have been used to measure nonlinearly induced phase distortion (Adair et al., 1989; Weber et al., 1978; Moran et al., 1975; Xuan et al., 1984). Often a sample is placed in one path (e.g., arm) of an interferometer, and the interference fringes are monitored as a function of irradiance. For example, if the interferometer is first set to give a series of straight-line interference fringes for low input (linear regime), the fringes become curves at high inputs near the central, high-irradiance portion of the beam. The addition of a streak camera can add time resolution to the analysis (Moran et al., 1975). Alternatively, a third beam pathway can be added so that fringes from two weak beams are monitored and the sample is in the path of one weak beam and the strong third beam. Then the fringe shift occurs when the strong beam is blocked and unblocked, giving the optical path-length change from which the phase shift

300

MANSOORSHEIK-BAHAE AND ERICW.VAN STRYLANLI

can be determined (Xhan et al., 1984). Jnterferometricmethods require good stability and precise alignment; however, such techniques using various modulation schemes have resulted in sensitivities of better than A/104in the induced optical path-length changes. 6. Z-SCAN Z-scan was developed for measuring nonlinear refraction (NLR) and determining its sign (Sheik-Bahae et al., 1989). It was soon realized that it also was useful for measuring nonlinear absorption (NLA) and separating the effects of NLR from NLA (Sheik-Bahae et al., 1990). We start by explaining its use for determining NLR. Using a single focused beam, as depicted in Fig. 18,we measure the transmittance of a sample through an aperture (Z-scan) or around an obscuration disk (EZ-scan) (Xia et al., 1994; Van Stryland et al., 1994), where either are positioned in the far field. The transmittance is determined as a function of the sample position Z measured with respect to the focal plane. Using a Gaussian spatial profile beam simplifies the analysis. The following example qualitatively describes how such data (2-scan or EZ-scan) are related to the NLR of the sample. Assume, for example, a material with a positive nonlinear refractive index.

I1

Detector Sample

FIG. 18. Z-scan geometry with reference detector to minimize background and maximize the signal-to-noise ratio. Figure adapted from Sheik-Bahae et al. (1989).

4 OPTICAL NONLINEAR~TIES IN BULK SEMICONDUCTORS

’

0.97 -5.0

-2.5

0

2.5

301

5.0

212,

FIG. 19. Predicted Z-scan signal for positive (solid line) and negative (dashed line) nonlinear phase shifts.

Starting the Z-scan (i.e., aperture) from a distance far away from the focus (negative Z), the beam irradiance is low, and negligible NLR occurs; hence the transmittance remains relatively constant. The transmittance here is normalized to unity, as shown in Fig. 19. As the sample is brought closer to focus, the beam irradiance increases, leading to self-focusing in the sample. This positive NLR moves the focal point closer to the lens, leading to a larger divergence in the far field. Thus the aperture transmittance is reduced. Moving the sample to behind the focus, the self-focusing helps to collimate the beam, increasing the transmittance of the aperture. Scanning the sample further toward the detector returns the normalized transmittance to unity. Thus the valley followed by peak signal is indicative of positive NLR, whereas a peak followed by a valley shows self-defocusing.Figure 19 shows the expected result for both negative and positive self-lensing. The EZ-scan reverses the peak and valley because, in the far field, the largest fractional changes in irradiance occur in the wings of a Gaussian beam. The EZ-scan can be more than an order-of-magnitude more sensitive than the Z-scan. We can define an easily measurable quantity ATp as the difference between the normalized peak and valley transmittance: T, - T,. The variation of ATp is found to be linearly dependent on the temporally averaged induced phase distortion, defined here as Amo [for a bound-electronic n2, AQ0 involves a temporal integral of Eq. (61) without carrier refraction, i.e., cr, = 01 (Sheik-Bahae et af., 199Ob). For example, in a Z-scan using a small

302

MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

aperture with a transmittance of S < lo%,

assuming CW illumination. With experimental apparatus and data-acquisition systems capable of resolving transmission changes ATpvz 1%, Z-scan is sensitive to less than ;1/250 wavefront distortion (i.e., Amo = 2n/250). The Z-scan has a demonstrated sensitivity to a nonlinearly induced optical path-length change of nearly A/103, whereas the EZ-scan has shown a sensitivity of /./lo4, including temporal averaging over the pulse width. Here the temporal averaging for an instantaneous nonlinearity and Gaussian temporal shape gives AOo = AQpeak/& whereas for a long-lived nonlinearity (much longer than the pulse width), AQo = AO/2 independent of the pulse shape. In the preceding picture we assumed a purely refractive nonlinearity with no absorptive nonlinearities such as 2PA that will suppress the peak and enhance the valley. If NLA and NLR are present simultaneously, a numerical fit to the data can in principle extract both the nonlinear refractive and absorptive coefficients. However, a second Z-scan with the aperture removed and care taken to collect all the transmitted light can determine the NLA independently. For 2PA alone and a Gaussian input beam, the loss nearly follows the symmetric Lorentzian shape as a function of the sample position 2. The magnitude of the loss determines the NLA, e.g., fi from Eq. (71). This so-called open aperture Z-scan is only sensitive to NLA. A further division of the apertured Z-scan data (referred to as closed-aperture Z-scan) by the open-aperture Z-scan data gives a curve that for small nonlinearities is purely refractive in nature (Sheik-Bahae et al., 1990b). In this way we have separate measurements of the absorptive and refractive nonlinearities without the need for computer fits of the 2-scans. Figure 20 shows such a set of Z-scans for ZnSe. Here the lines are numerical fits to the curves. Separation of these effects without numerical fitting for the EZ-scan is more complicated. 7. EXCITE-PROBE Z-SCAN Excite-probe techniques in nonlinear optics ..ave been employed to deduce information that is not accessible with a single-beam geometry (Shank et al., 1978). By using two collinear beams in a Z-scan geometry, we can measure nondegenerate nonlinearities, we can temporally resolve these nonlinearities, and we can separate the absorptive and refractive contributions. There have been several investigations that have used Z-scan in an

4 OPTICAL NONLINEARITIES IN BULKSEMICONDUCTORS

0 0

303

1.04

5

=

4

Closed

0.90

E :5:om B

:om

g

= 0.72

Division (index)

FIG. 20. Z-scans for ZnSe using picosecond 532-nm pulses: (a) open aperture; (b) closed aperture; (c) closed aperture data divided by open aperture data. Figure adapted from Sheik-Bahae er a/. (1991).

304

MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

excite-probe scheme. Z-scan can be modified to give nondegenerate nonlinearities by focusing two collinear beams of different frequencies into the material and monitoring only one of the frequencies (different polarizations can be used for degenerate frequencies) (Ma et al., 1991; Sheik-Bahae et al., 1992). The general geometry is shown in Fig. 21. After propagation through the sample, the probe beam is then separated and analyzed through the far-field aperture. Due to collinear propagation of the excitation and probe beams, we are able to separate them only if they differ in wavelength or polarization. The former scheme, known as a two-color Z-scan, has been used to measure the nondegenerate n, and #? in semiconductors. Figure 22 shows results of such experiments performed on ZnSe and ZnS samples with excitation at 1.06pm and probing at 532nm, i.e., p ( 2 q o)and n,(2o; o) (Sheik-Bahae et al., 1994). The data are scaled as before (see Eqs. 51 and 5 5 ) and plotted to show comparison with the TPB model for FzpAand G,. The most significant application of excite-probe techniques in the past concerned the ultrafast dynamics of nonlinear optical phenomena. The two-color Z-scan can separately monitor the temporal dynamics of NLR and NLA by introducing a temporal delay in the path of one of the input beams. These time-resolved studies can be performed in two fashions. In one scheme, Z-scans are performed at various fixed delays between excitation and probe pulses. In the second scheme, the sample position is fixed (e.g., at the peak or the valley position), while the transmittance of the probe is measured as the delay between the two pulses is varied. Figure 23 shows the result of using this second method on ZnSe to separately determine the dynamics of the NLA and NLR (i.e., the time-dependent signal at the valley is subtracted from that at the peak) [Wang et al., 19943. The analysis of

filter

FIG. 21. Optical geometry for a two-color 2-scan. The filter blocks the pump beam. Adapted from Sheik-Bahae er 01. (1992).

4 OPTICAL NONLINEARITIES IN BULKSEMICONDUCTORS

305

0.10 n

a

Q

fi0.05

0.00 0.10

0.00

3 -0.10 I

0.0

.

I

0.2

.

I

0.4

.

I

0.6

.

I

0.8

.

I

1.0

W E , FIG.22. (a) The measured degenerate j(2w; 2 0 ) (open symbols) and nondegenerate j(2w; o)(solid symbols) for ZnSe (circles) and ZnS (triangles) using w = 2 m / I with I = 1.06pm. The data are scaled according to Eq. (51) to compare with the TPB theory; F2(2x; x) (solid line) and F,(x; x ) dashed line. (b) The corresponding measured n2 values scaled according to Eq. (55) to compare with the TPB theory; G,(2x; x ) (solid line) and G,(x,x) (dashed line). Figure from Sheik-Bahae et al. (1992).

two-color Z-scans is naturally more involved than that of a single-beam Z-scan. The measured signal, in addition to being dependent on the parameters discussed for the single-beam geometry, also will depend on parameters such as the excite-probe beam waist ratio, pulse-width ratio, and the possible focal separation due to chromatic aberration of the lens (Wang et al., 1994; Ma et af., 1991; Sheik-Bahae et al., 1992). Table 111 gives the results of data for ZnSe taken using transmittance, beam distortion, Z-scan, two-color Z-scan, and time-resolved excite-probe techniques. Another excite-probe technique based on Z-scan geometry is the method of Kerr-lens autocorrelation (Sheik-Bahae, 1997; Sheik-Bahae and Ebrahimzadeh, 1997) suitable for measurements employing femtosecond laser pulses.

MANSOORSHEIK-BAHAE AND

306 0.05 I

ERIC 1.1

I P)

W.

-

V A N STRYLAND

,

7,=1ns

0

c

c

n 0

-

-0'23200 -100

.0

200

100

300 400

z0'1200 -1'00

Time Deioy (ps)

6

160

60 360 400

Time Delay '(ps)

FIG.23. Time-resolved Z-scan data on ZnSe using 532-nm, picosecond excitation pulses and probing at 1.06pn: (a) nonlinear refraction versus temporal delay; (b) nonlinear absorption versus temporal delay. Figure from Wang et a/. (1994).

TABLE 111 DATA TAKEN ON CVD-GROWN ZNSEAND ZNS USING SEVERAL OF THE TECHNIQUES DISCUSSED I N THISSECTION

ZnSe

ZnS

b( 1.06; 1.06)

0

p(0.532; 0.532) p(0.532; 1.06) /3( I .06; 0.532) py( 1.06;0.532) n2(1.06;1.06) ~ ~ ( 0 . 5 30.532) 2; ti2(0.532;1.06) n:'(0.532; I .06) n2( 1.W, 0.532) ( i . ( 1.06) fl,( 1.06)

5.8 f 1 cm/GW

0 3.4 f 0.7 cm/GW 0 0 0 (6.3 k 1.4) x 10-'5cm2/W Not measured (1.7 k0.4) x 10-14cm2/W Not measured < 1.5 x 10-14Cm2/w (7 2) x 10-l8cm2 (5.2 & 1 . 1 ) x 1 0 - 2 2 ~ m 3 re z 0.6 ns; T, = 0.8 ns

L

I5 k 3cm/GW 4.6 f I cm/GW 8.6 2 cm/G W (2.9 f 0.3) x 10-'4cm2/W (-6.8 +_ 1.4) x 10-14cm/W (-5.1 k0.5)x 10-15cmZ/W (-2.6 kO.3) x 10-'4cm2/W (-9 f 5) x 1 0 - ~ 5 ~ m 2 / w (4.4 f 1.3) x 1O-'*cm2 (-6.1 & 1.5) x IO-''cm3

-

s

Note: The x y superscript indicates that the two beams in the two-color Z-scan were perpendicularly polarized. The T~ and 7, for ZnS indicate that the decays seen in the time-resolved two-color Z-scan for absorption and refraction were different.

8. FEMTOSECOND CONTINUUM PROBE

The development of high-irradiance,femtosecond pulsed laser systems has allowed a pump/probe experiment that automatically yields the non-

4 OPTICAL NONLINEAR~TIES IN BULKSEMICONDUCTORS

307

degenerate nonlinear absorption spectrum. In such an experiment, the femtosecond pulse is split in two, and one beam is used as the excitation, while the other beam is focused into a suitable material to produce a white-light continuum (Bolger et al., 1993). This white-light continuum is then used as the probe at all frequencies orin Eq. (29). Given a sufficiently broad spectrum, the KK integral can be applied to yield the nondegenerate n,. This method has not been applied to date over a broad spectral range. However, in principle, both nondegenerate B and n, can be obtained in a single shot measurement. 9. INTERPRETATION

The interpretation of NLA and NLR measurements is fraught with many pitfalls and great care must be taken. In extensive studies of a wide variety of materials, it is found that there is seldom a single nonlinear proces occurring. Often several processes occur simultaneously, sometimes in unison and sometimes competing. It is necessary to experimentally distinguish and separate these processes in order to understand and model the interaction. There are a variety of methods and techniques for determining the nonlinear optical response, each with its own weaknesses and advantages. In general, it is advisable to use as many complementary techniques as possible over as broad a spectral range as possible to unambiguously determine the active nonlinearities. Numerous techniques are known for measurements of NLR and NLA in condensed matter, including the methods discussed earlier. Nonlinear interferometry (LaGasse et al., 1990; Weber et al., 1978; Moran et al., 1975; Xuan et al., 1984), degenerate four-wave mixing (DFWM) (Canto-Said et al., 1991; Fisher, 1983), nearly degenerate three-wave mixing (Adair et al., 1989), ellipse rotation (Owyoung, 1973), beam distortion (Williams, 1984), beam deflection (Bertolotti, 1988), and third-harmonic generation (Kajzar and Messier, 1987) are among the techniques frequently reported for direct or indirect determination of NLR. Z-scan is capable of separately measuring NLA and NLR (Sheik-Bahae et al., 1989, 1990b). Other techniques for measuring NLA include transmittance (Bechtel and Smith, 1976), calorimetry (Bass et al., 1979), photoacoustic (Bae et al., 1982; Van Stryland et al., 1980), and excite-probe (Shank et al., 1978) methods. VII. Applications Ultrafast nonlinearities in optical solids have been used for applications ranging from ultrashort laser pulse generation (Kerr-lens mode locking) to

308

MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

soliton propagation in fibers over distances of the earth’s circumference. Here we briefly discuss two areas of research using optical nonlinearities in the transparency region of semiconductors: (1) all-optical switching, a potential device application in telecommunications switching and routing systems, and (2) optical limiting, primarily applicable to protecting optical sensors from high-irradiance inputs.

1. ULTRAFASTALL-OPTICAL SWITCHING USINGBOUND-ELECTRONIC NONLINEARITIES

An important application of the n2-jtheory that was presented in Section 111 is that it allows direct determination of the ideal operating point of a

passive optical switch. Optical switch designers have established a figure of merit (FOM) for candidate materials, defined by the ratio nn,//?L = 1/T (where T is the FOM defined in Mizrahi er al., 1984). The goal of maximizing the FOM clearly shows the need for a large nonlinear phase shift (nnJ.2) while keeping the 2PA loss (B) small. By substituting Eqs. (48) and (52) in the FOM ratio, the theory can be used to obtain a universal FOM curve. This FOM is then given by xG2(x)/F2(x)= 1/T, where x = hO/EP. Figure 23 depicts the calculated FOM and a comparison with experimental data obtained for several semiconductors (Sheik-Bahae er al., 1991). The remarkable agreement between theory and experiment indicates that this quantity is indeed a fundamental property of semiconductors, depending only on the normalized optical frequency (ho/E,). The two horizontal lines in Fig. 24 represent the minimum acceptable FOM for nonlinear directional couplers (NLDC) and Fabry-Perot (FP) interferometers. Although it demands a larger FOM, the NLDC scheme is the preferred practical geometry. From Fig. 24 we see that the FOM requirement is satisfied either just below the 2PA edge or very near resonance (hw 5 E,). Since n2 a Eg-4, a low switching threshold at a given wavelength demands a material with the smallest possible band-gap energy. The theory then suggests that the ideal operating region is just below the band gap. However, operation near the band gap forces the designer to contend with increasing loss due to band-tail linear absorption, which makes this scheme unworkable at present (at least in passive material). If operation near the half band gap is contemplated instead, one must pay the penalty of reduced nonlinear refraction (- 16 times at a given wavelength). To compensate, the operating irradiance must be increased. At high irradiance, however, nonlinear absorption associated with 2PA becomes an issue, making this option problematic as well. Therefore, passive all-optical switching presents fundamental constraints that cannot readily be solved by

4 OPTICAL NONLINEARITE IN BULK SEMICONDUCTORS

309

AdE, FIG. 24. Ratio of n,/k/? (switching parameter or figure of merit) as a function of ho/E,. The solid line is predicted from the two-parabolic-band model. NLDC stands for nonlinear directional coupler, and FP stands for Fabry-Perot etalon. Figure from Sheik-Bahae et al. (1991).

materials engineering. One method being pursued is operating just below the 2PA edge so that 3PA is the dominant loss mechanism. This has led to some promising results (Stegeman et al., 1996). Another method, as discussed previously, is using semiconductor laser amplifiers (SLA), where parasitic linear loss can be mitigated, making near-gap operation a practical possibility. 2. OPTICAL LIMITING Passive optical limiting uses a material’s nonlinear response to block the transmittance of high-irradiance light while allowing low-irradiance light to be transmitted (an operation similar to that of photochromic sunglasses). The primary application of optical limiting is to protect sensitive optical components from being damaged by the high-intensity input light. The ideal optical limiter has a high linear transmission for low inputs (e.g., energy), a variable limiting input energy, and a large dynamic range defined as the ratio of the linear transmittance to the minimum transmittance obtained for high input (prior to irreversible damage) (Crane et af., 1995; Sutherland et al., 1997). Since a primary application of optical limiting is to protect sensors, and fluence (energy per unit area) almost always determines

310

MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

damage to detectors; this is the quantity of interest for the output of a limiter. Getting this type of response is possible using a wide variety of materials; however, it is very difficult to get the limiting threshold as low as is often required and at the same time have a large dynamic range. Because high transmission for low inputs is desired, we must have low linear absorption. These criteria lead to the use of materials displaying strong 2PA and nonlinear refraction. Devices based on these nonlinearities can be made to have low limiting thresholds, large dynamic ranges, and broad spectral responses; however, since 2PA and n2 are irradiance-independent, they work best for short input pulses (Hagan et al., 1988). For example, a monolithic ZnSe device limits the output fluence at input energies as low as lOnJ (300W peak power) and has a dynamic range greater than lo4 for 532-nm, 30-ps (FWHM) pulses, as shown in Fig. 25 (Van Stryland et al., 1988). While the nonlinear response of this device is initiated by 2PA, free-carrier defocusing greatly assists the limiting of the transmitted fluence and is responsible for increasing the dynamic range over which semiconductor limiting devices operate without damage. Since the light is focused in the bulk of the material (see inset of Fig. 25), the semiconductor could itself be damaged. However, at high inputs, the combination of 2PA loss and carrier defocusing that counteracts linear focusing protects the focal position from damage. For longer-pulse operation, however, the dynamic range is significantly reduced. This occurs because the energy input for the same irradiance

--z

2.2r-

. .. ... :... .. .. . .. - .. *. . . . *'*. .. " .*:. .'.

C

3

2.0

B2

1.a

' w

-

I..

-

,

1.6-

0

*:

1.4-

3

1.2

u.

-

0.0

g

0.6

f

g

-

--u-

-*-

..

'58*.'.**..'-:..-..'.:

-.

.

0.4

0.2

:

0.0

*

:m--

! ! 1.05

g

*

c I

A n

I

___----

_/--

-.-----_

I

FIG. 25. Optical limiting data for the monolithic ZnSe lens ( - 3 an long) shown in the inset for picosecond, 532-nm pulses as a function of the input energy as measured though an aperture approximately 2.5m after the sample. Adapted from Van Stryland et al. (1988).

4 OPTICAL NONLINEARITIES IN BULKSEMICONDUCTORS

31 1

is increased for longer pulses. While more carriers are generated and free-carrier absorption also becomes significant, they may decay during the pulse, and the energy from nonlinear absorption heats the bulk of the sample. This heat raises the refractive index in most semiconductors. The increase in refractive index causes self-focusing that counteracts the freecarrier defocusing and the sample damages. As seen from the 2PA scaling relations, the 2PA can be greatly enhanced for infrared wavelengths where smaller band-gap energies can be used (fl oc EB3). For example, InSb at 10pm has B s lo4 cm/GW, and the free-carrier absorption and refraction are very large, dominating the nonlinear response. This material has great potential for sensor protection in the IR (Hasselbeck et al., 1993).

VIII. Conclusion Since the advent of high-peak-power short-pulse lasers, numerous measurements of the ultrafast optical Kerr effect (n,) in many semiconductors and large-gap dielectrics have been reported. The experimental techniques used for these measurements range from nonlinear wave mixing to nonlinear interferometry. Almost all the early measurements were obtained in the long-wavelength limit, where n2 is positive and nondispersive. More recent measurements have shown the dispersive nature of the nonlinear refraction (Sheik-Bahae et al., 1991). A simple two-band model calculation gives a universal band-gap scaling and dispersion of the electronic Kerr effect in solids. A direct relationship links the nonlinear refractive index n, to its nonlinear absorptive counterparts: two-photon absorption, Raman, and AC Stark effects. This theory builds from a large base of existing calculations where nonlinear absorption is calculated by means of transition rates. An appropriate Kramers-Kronig transformation approach is used to obtain the nonlinear refraction in terms of this nonlinear absorption. The power of this approach is that it circumvents the need for a direct calculation of the complex nonlinear susceptibility. It is necessary, however, to know the nondegenerate nonlinear absorption coefficient in order to apply the Kramers-Kronig transformation, i.e., the nonlinear absorption in the material at all frequencies w , in the presence of a strong optical field at 0,.The n, calculation is also performed for the general nondegenerate case where an expression for n,(w,; 0,)is derived. This is the coefficient of nonlinear refractive index at w1 due to the presence of a strong optical excitation at 0,.The well-known and wellstudied degenerate n2 is treated as a special case. Comparing the experimentally measured values of the degenerate n, with the theoretical dispersion,

312

MANSOORSHEIK-BAHAE AND ERICW. VAN STRYLAND

there is good agreement obtained over a wide range of frequencies and materials with a single fitting parameter K‘. We note, however, that the theoretical value for this parameter is a factor of 2 to 3 smaller than the empirical value of K’. This underestimation may be expected because the heavy-hole valence band, as well as the electron-hole Coulomb (exciton) interaction, is ignored in this simple theory (Sheik-Bahae et al., 1994). It is also remarkable that the two-band theory gives reasonable agreement with data for large-gap dielectric materials. The theory for passive semiconductors also can be extended to active semiconductor devices, semiconductor laser amplifiers (SLA). The measured sign and magnitude of n,, as well as the variation of n, with injection current density in SLA systems, is in good agreement with calculations. While the ultrafast nonlinearities of semiconductors can now be predicted with reasonable accuracy given the band-gap energy, linear index, and photon energy, other nonlinearities are often important for device applications. In particular, free-carrier and thermal nonlinearities can significantly alter the nonlinear operation. In practice, the shorter the input pulse, the less these nonlinearities interfere with the simple modeling of the ultrafast response. This occurs because the shorter the pulse, the less energy for a given irradiance and, therefore, the fewer carriers are produced and the less heat is generated. We end our discussion with a reminder of the difficulties in characterizing the nonlinear optical properties of materials and in particular semiconductors. For example, for photon energies near the band edge, there can be significant linear absorption, and this linear absorption leads to the creation of free carriers that can subsequently absorb and refract light. The refractive component is the more interesting for applications and this resonant nonlinear refraction gives one of the largest nonlinearities ever reported (Miller and Duncan, 1987). However, it and the associated NLA can interfere with the determination of either two-photon absorption or n2 (we restricted our definition of n, to the ultrafast optical Kerr effect from the bound electrons). Without knowledge of the temporal dynamics, both nonlinearities result in a third-order response. In a similar fashion, nonlinear absorption and nonlinear refraction from 2PA-generated carriers result in a fifth-order nonlinearity that is difficult to distinguish from three-photon absorption and “n3,” the fifth-order bound-electronic nonlinear refraction. In all of this, of course, are the problems associated with the interactions between nonlinear loss and nonlinear refraction with multiple sources of nonlinearities; e.g., nonlinear absorption leads to beam-profile changes that alter the propagation, and nonlinear refraction through propagation alters the beam profile. I n short, great care must be taken to determine the underlying physics associated with nonlinearities in semiconductors (and other materials).

4 OFTICALNONLINEARITIES IN BULKSEMICONDUCTORS

313

ACKNOWLEDGMENTS We gratefully acknowledge the contributions of Drs. D. J. Hagan and D. C. Hutchings. EVS also thanks a number of former students and postdoctoral researchers. The financial support of the National Science Foundation is greatly appreciated over the past several years, as is support from the Joint Services Agile Program.

L I OF~ ABBREVIATIONS AND ACRONYMS 2PA AOS BF

cw

DFWM FCA FCR FP FOM

two photon absorption all-optical switching band-filling continuous wave degenerate four-wave mixing free-carrier absorption free-carrier refraction Fabry-Perot figure of merit

hh KK Ih-c NLA NLDC NLR

QSE SLA TPB

heavy-hole Kramers-Kronig light-hole-to-conduction band nonlinear absorption nonlinear directional couplers nonlinear refraction quadratic optical Stark effect semiconductor laser amplifier two-parabolic band

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SEMICONDUCTORS AND SEMIMETAI S VOL.58

CHAPTER5

Photorefractivity in Semiconductors James E. Millerd M m o LASOI. INC. IRVMS CALIFURN&

Mehrdad Ziari SDL, INC. SAN JW CALIFORNU

Afshin Partovi BELLLAWF~ATOR~~S LUCENT TECHN~~IES MURRAYHILLNew JERSEY

1. INTRODUCTION . . . . . . . . . . . . . . . . . 11. SPACE-CHARGE GRATING FORMAT'ION . . . . . . . .

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320 321 321 323 324 326 4. Linear Electro-Optic Effect . . . . . . . . . . . . . . . . . . . . . . 5. Other Dielectric Modulation Mechanisms . . . . . . . . . . . . . . . . 327 I11. BEAM COUPLING . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 1 . Coupled-Wave Equations . . . . . . . . . . . . . . . . . . . . . . . 328 2. Spatial Frequency Dependence . . . . . . . . . . . . . . . . . . . . . 330 3. Intensity Dependence . . . . . . . . . . . . . . . . . . . . . . . . . 332 4. Temporal Response . . . . . . . . . . . . . . . . . . . . . . . . . 333 5. Electron-Hole and Multidefect Interactions . . . . . . . . . . . . . . . 336 IV. FOUR-WAVE MIXING. . . . . . . . . . . . . . . . . . . . . . . . . . 337 338 1 . Degenerate Four- Wave Mixing . . . . . . . . . . . . . . . . . . . . 2. Dijiraction Eficiency Measurements . . . . . . . . . . . . . . . . . . 340 342 3. Self-pumped Phase Conjugation . . . . . . . . . . . . . . . . . . . . 4. Polarization Switching . . . . . . . . . . . . . . . . . . . . . . . . 344 v. ENHANCED WAVE-MIXING TECHNIQUES . . . . . . . . . . . . . . . . . . 346 346 1. DC Applied Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 2. ACFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 350 3 . Moving Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . 1. Plane- Wave Interference Model . . . . . . . . . . . . . . . . . . . . 2. Simplified Band Transport Model . . . . . . . . . . . . . . . . . . . 3. Steady-State Solution . . . . . . . . . . . . . . . . . . . . . . . .

List of Abbreviations and Acronyms can be located preceding the references to this chapter.

319 Copyright (0 1999 by Academic Press All rights or reproduction in any form reserved. ISBN 0-12-752167-4 ISSN 0080-8784/99 130.00

320

JAMB E. MILLERD.MEHRDADZIARIAND AFSH~NPARTOVI 4. Temperature-Intensity Resonance

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5 . Near-Band-Edge Effects . . . . . . . . . . . . . . . . . . . . . 6. Photorefractive Response at High Moablation Depths . . . . . . . . . 1. Summary of Applied Field Techniques . . . . . . . . . . . . . . . . VI . BULKSEMICONDUCTORS . . . . . . . . . . . . . . . . . . . . . . . 1 . &As . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . InP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.GuP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.CdTe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.ZnTe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. CdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . Bulk 11-VI All0.v~ . . . . . . . . . . . . . . . . . . . . . . . . V I I . MULTIPLE QUANTUM WELLS. . . . . . . . . . . . . . . . . . . . . 1. M Q W-PR Devices Using the Quantum Conjned Stark Efect . . . . . . 2 . Elimination of the Deposited Layers and Substrate Removal in PR-MQ Ws 3. M Q W-PR Using the Franz-Keldysh Effect . . . . . . . . . . . . . . vl11. SELECTED APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . 1 . Coherent Signal Detection (Adaptive Inter$erometer) . . . . . . . . . 2 . Optical Imuge Processing . . . . . . . . . . . . . . . . . . . . . 3. Optical Correiators . . . . . . . . . . . . . . . . . . . . . . . 4. Real-Time Holographic Interferometry . . . . . . . . . . . . . . . LIST OF ABBREVIATIONS AND ACRONYMS. . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . .

.. . . . . . .

. . . . . .

352 354 361 363 364 365 366 367 367 361 369 369 371 373 380 383 385 385 387 388 391 394 395

Introduction

The photorefractive effect in semiconductors leads to nonlinear interactions between optical waves such as four-wave mixing; however. the required intensity levels are orders of magnitude less than those necessary for true high-order nonlinearities (e.g., x 3 ) because they rely on carrier transport (see Chap. 2). Photorefractive gratings arise from linear absorption. transport and storage of charge. and the linear electro-optic effect (Nolte. 1995). The magnitude of a photorefractive grating is determined by spatial intensity gradients. while the speed of formation is a function of overall intensity. Because it is present at very low intensity levels. the photorefractive effect is useful to the material scientist as a tool to study charge transport and defect levels in materials. and it is useful to the electro-optic engineer as a method for real-time optical wavefront processing. phase conjugation. beam combining, and holographic interferometry. The first section of this chapter will present the fundamentals of photorefractive grating formation by considering sinusoidal illumination (i.e., two beams writing a grating in the volume of the semiconductor crystal). The mechanisms of space-charge grating formation and electro-optic coupling that enable the space-charge field to interact with the optical fields will be

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS

321

presented. The second section will cover the phenomenon of beam coupling, where a spatially shifted grating gives rise to unidirectional energy transfer between the beams that produced the grating. The third section addresses the case of four-wave interaction where a phase conjugate beam is generated. The magnitude of the photorefractive effect can be enhanced by increasing the space-charge field through the application of external electric fields, as well as by using higher-order electro-optic mechanisms. The methods, limitations, and nonlinearities that occur when using external fields and higher-order electro-optic effects to enhance the photorefractive gain coefficient are presented in the fourth section. The fifth section presents a review of the different bulk photorefractive semiconductor compounds that have been investigated. The sixth section reviews photorefractive multiple-quantum-well structures. The final section presents some of the promising applications that are being explored for photorefractive semiconductors.

11. Space-Charge Grating Formation 1. PLANE-WAVEINTERFERENCE MODEL The photorefractive effect arises from a spatially nonuniform intensity pattern. The simplest way to understand the photorefractive process is to consider the case of two plane waves interfering inside a semiconductor crystal. Figure 1 shows the basic arrangement that will be used to illustrate photorefractive grating formation. The process happens as follows: (1) a sinusoidal intensity pattern is produced inside the crystal as a result of the interference of the two plane waves, (2) light is absorbed and free carriers are generated in the bright regions of the intensity pattern (in the example of Fig. 1, electrons are the free carriers), (3) the carriers diffuse and/or drift from the bright regions, leaving fixed charges behind, (4) the carriers are trapped at deep levels in the dark regions (due to point defects that may be intentionally introduced), ( 5 ) the resulting nonuniform charge distribution causes a spatially varying electric field or space-charge field, and (6) the space-charge field modulates the refractive index of the crystal through the electro-optic effect. The refractive index grating inside the crystal can be “read out” by diffracting a third beam from it, or depending on the polarization of the writing beams, crystal orientation, and spatial phase of the grating, the writing beams may diffract from it themselves. The key point here is that the spatially nonuniform intensity pattern leads to a nonuniform charge distribution and finally to a nonuniform refractive index distribution inside the semiconductor crystal.

322

JAMES E. MILLERD,MEHRDADZIARIAND AFSHINPARTOVI X

Interference pattern FIG. I .

Space chargefield

Basic arrangement for writing a photorefractive grating in a crystal.

The incident intensity pattern caused by the two beams interfering inside the volume of the crystal can be written as I(x) = I,[1

+ mcos(k,x)]

(1)

where I, is the combined intensity of the pump (or reference) and signal beams, I, = I, + fS, k, = 2n/A is the grating vector, A = A/(2 sin 8) is grating period, i. is the free-space optical wavelength, 8 is the half angle between beams measured outside the crystal, and

is the modulation index. Here it has been assumed that the two waves are both polarized perpendicular to the plane of incidence. For arbitrary polarization states, the modulation index is reduced by the dot product of the polarization unit vectors ( e g , for orthogonally polarized beams rn = 0). This spatial intensity gradient is the driving function for space-charge grating formation. Although the plane-wave model is simplified, its relevance can be appreciated by noting that arbitrary wavefronts (e.g., from an aberrated beam or image) can be treated as a superposition of many plane waves. In addition to the transmission grating shown in Fig. 1, reflection gratings also can be written in photorefractive semiconductors. For a reflection configuration, the two beams are incident from opposite sides of the crystal.

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS

323

In this case, the period of the grating is A = R/2n sin 8',where n is the index of refraction of the semiconductor and 6' is the half angle between the beams inside the crystal. For large angles (e.g., counterpropagating beams), the grating spacing can be smaller than the free-space wavelength.

2. SIMPLIFIED BANDTRANSPORT MODEL The incorporation of point defects in the semiconductor is imperative for the photorefractive effect. The point defects are responsible for photoconductivity at photon energies below the bandgap and permit the buildup of a space-charge field through the spatial trapping of charge. Point defects can occur in undoped material, as in the case of GaAs:EL2, or can be intentionally doped, as in InPFe. A simplified energy-level diagram for the band structure in a photorefractive crystal is shown in Fig. 2. In this model, a deep donor level is partially compensated by an acceptor level. At room temperature, the acceptor level is completely full and the deep level donor is partially full, pinning the Fermi-level to midgap. On absorption of light from nonuniform illumination, electrons are promoted from the deep donor level to the conduction band, where they drift and/or diffuse into the darker regions and recombine with a deep level. Exposure to uniform light returns the crystal to a spatially charged neutral condition. The relevant equations for this process are (Gunter and Huignard, 1987) (Rate equation) (The continuity equation)

dN2 dt

+ BMND - N ; ) dn d N 2 1 - _ --- + - V . j -- - (sZ dt

dt

- y,nND'

e

(The current equation) (Poisson's equation)

(3) (4)

(5)

V-EE, = -e(n

+N,

- N,+)

(6)

where n is the electro density, N , is the concentration of acceptors, N D is the concentration of deep-level donors, N A is the concentration of ionized donors, pe is the electron mobility, e is the electron charge, K, the Boltzman constant, T is temperature,j is current, E is the dielectric constant ( E = E , E ~ ) , I is incident intensity, s is the optical absorption cross section, fi is the thermal ionization rate of the donors, y, = 1 / ~ ,is the recombination coefficient, rr is the carrier lifetime, and E,, is the space charge field inside the material.

324

JAMES E. MILLERD,MEHRDADZIARIAND AFSHINPARTOVI

-

+ - +

W m )

FIG.2. Simplified band-transport model for a photorefractive crystal with a partially occupied deep level.

Equation (3) is the basic rate equation that incorporates the optical and thermal generation of charged carriers (in this case, electrons). The continuity equation, Eq. (4), accounts for charge motion in the rate equation. Equation (5) accounts for both drifts in response to electric fields and diffusion of charged carriers. Poisson's equation relates the electric field to the charge distribution. These four equations govern the buildup of photorefractive gratings; however, the simplified band-transport model is inadequate for predicting the behavior of many real materials (Nolte et al., 1989a; Delaye et al., 1990). Coupling to the valence band and multiple defect levels often complicate the dynamics of the photorefractive process; nevertheless, the simple model does provide a useful physical picture, and in some cases, more complicated models can be accounted for by empirically modifying the solutions of the simple model. Section 111.5 examines some of the effects that occur when both holes and electrons are involved in optical excitation. 3. STEADY-STATE SOLUTION

The equations can be solved analytically by assuming that under steadystate conditions the modulation index is much less than unity, corresponding to a small signal analysis, and by only considering the zero- and first-order harmonics (with respect to the periodicity of the incident intensity pattern) for the electron density, ionized donor concentrations, and space-

5

P€IOl'ORFFRACTlVITY IN SEMICONDUCIORS

325

charge fields. In this simplified model it is typically assumed that the concentration of donors and acceptors are such that n << N A << N D . Equations (1) to (6) are combined, and for a steady-state solution the time derivatives are set equal to zero (i.e., no net charge buildup in the steady state). The space-charge field generated inside the material in response to the steady-state optical intensity pattern can be written as

E, = E,

+ lEll cos(k,x + 4)

where E , is the zero-order electric field (usually equal to the applied field), and E , is the first-order coefficient of the space-charge field. The phase 4 is introduced to include the fact that the space-charge grating can be offset laterally from the interference pattern. It will be seen later that it is necessary to include higher-order harmonics when using large applied fields at high modulation index. In general, a numerical method is necessary to solve for highest-order terms of the coupled differential equations; however, firstorder analysis is adequate for a surprising number of situations. If the optical generation rate is much greater than the thermal generation rate, a1 >> 8, then the steady-state solutions are independent of intensity and are given by

(9)

where 2nk,T ED =Ae

N,Ae

E,, = 2AE

E , is the externally applied electric field, and N,is the effective trap density in the material (in the case of the simple band model, N , = NA).ED is known as the diffusion-limited field and represents the maximum space-charge field that may build up due to diffusion of carriers from the bright spots. E, is the trap-limited field and represents the maximum field that may be produced if all the available charge inside each grating period is separated. The periodic electric field couples through the linear electro-optic effect to produce a refractive index grating that can diffract light.

326 4.

.fAMEs

E.

MILLFXD,MEHRDAD ZlARI AND

AFSHINPARTOVl

LINEARELECTRO-OPTIC EFFECT

The electro-optic effect is a change in the dielectric constant (or refractive index) of the material in response to electric fields. The tensor nature of the dielectric constant makes it possible for electric fields in one direction to affect light polarized in the same as well as orthogonal directions. The electro-optic effect is commonly expressed using the index ellipsoid (Yariv and Yeh, 1984)

A

($)

= 'i,jh

Eh

i.i

where ri.j,kis the electro-optic coefficient, E , is the electric field, and i, j, and k are each indexed along the fundamental crystal axis (x, y , and 2). The higher-order terms have been omitted from Eq. (11). In some situations, particularly near the band edge of the semiconductor, quadratic terms can be appreciable and even larger than the linear term. With a few exceptions, most photorefractive semiconductors belong to the 43m crystal symmetry class. There are only three nonzero electro-optic coefficients, r41 = rS2 = r63 for this symmetry class. The full index ellipsoid can then be written as

where no is the nominal refractive index of the crystal. For an electric field along an arbitrary direction, Eq. (12) can be recast to find the new principal axes and normal modes of propagation. That is, rearrange the equation into the form

where the primed values indicate the new principal axes. In general, this can be difficult to solve; however, we shall consider several orientations that are commonly used for photorefractive gratings. Figure 3 shows a common crystallographic cut (the so-called holographic cut) and two orientations for wave mixing. In the first case, the space-charge field is aligned along the 001 axis and along the 110 in the second case. For the first configuration, the change in refractive index is given by (Yariv, 1985)

An An

=

$&,,E

=0

for s-polarized light (along 110) for ppolarized light (along 001)

(14)

Thus the polarization of the writing beams can be adjusted to either experience the index grating or not. This polarization dependence is often

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS

327

FIG. 3. Two common configurationsand designations for two-beam coupling in semiconductors. Typically, the pump wave intensity I , is much stronger than the signal wave I,. Crystal configuration B can be used to suppress Pockels linear electro-optic beam coupling. (From Nolte, 1995.)

employed to isolate other dielectric modulation mechanisms (such as absorption or quadratic effects) from the linear electro-optic effect (Bylsma et al., 1988). For configuration B, the refractive index change is given by An = 0

for s-polarized light (along 110)

An = +n;r,,E

for p-polarized light (along 110)

(15)

which is the opposite of the preceding case. The maximum change in refractive index that can be obtained is achieved by writing and reading the grating along the 111 axis. In this case, the index change is 15% higher than the standard configurations of Eqs. (14) and (15) (Yariv, 1985). Because of the isotropic nature of cubic crystals and the tensor nature of the electrooptic effect, it is possible to couple energy between orthogonally polarized light beams. Cross-polarization coupling is discussed in Section IV.3.

5. OTHER DIELECTRIC MODULATIONMECHANISMS In addition to electro-optic gratings, other mechanisms exist that can modulate the refractive index of semiconductors in response to periodic illumination, including free-carrier or plasma gratings (Fabre et al., 1988;

328

JAMES E. MILLERD, MEHRDADZlARI AND

AFSHlN PARTOVI

Jarasiunas et al., 1993; Nolte et al., 1989b) and absorption gratings (Byslma et ai., 1988). In some instances, the magnitude of these gratings can exceed the electro-optic effect (Nolte, 1995). These other mechanisms are not covered in this chapter; however, the interested reader should consult the preceding chapter and the noted references.

Ill. Beam Coupling A unique feature of the photorefractive effect is that energy can be coupled unidirectionally between two beams. For example, a weak probe beam can be significantly amplified (2 to 3 orders of magnitude) by a strong pump beam, and rather amazingly, by simply changing the crystal orientation, the weak probe beam can be made to give up its energy to the stronger pump beam. Unidirectional energy transfer arises because of a spatial shift between the intensity grating and the refractive index grating. When the grating is shifted by one-quarter of a fringe period (90 degrees), unidirectional coupling is optimized.

1. COUPLED-WAVE EQUATIONS A coupled-wave analysis can be used to determine the energy transfer between the two beams (Kogelnik, 1969). The coupled-wave equations for two beams interfering inside a photorefractive crystal are

where A, and A, are the amplitude coefficients for the pump and signal waves, respectively, a is the linear absorption coefficient, and 4 is the phase of the space-charge grating. if we consider only the component of the grating that is spatially shifted by 90 degrees and substitute the expression for the change in refractive index, the coupled equations are reduced to d m -A =--FA dz 2 d m - A A , = - F2A P dz

a

--A 2 , a - p s

5

PH~TOREFRAC~IVITY IN SEMICONDUCTORS

329

where the gain coefficient

4nAn =-4nn~reff IE, I sin t$ Rmcos8' Rcos8' m

r=

lEll sin is the 90-degree spatially shifted component of the space charge field (= ImE,), and reff is the effective electro-optic coefficient, which accounts for crystal orientation and beam polarization. Equation (17) illustrates that the signal wave gains energy at the expense of the pump wave (for a positive gain coefficient). Changing crystal orientation or sign of the charged carriers can change the gain coefficient and hence the coupling direction. Typical gain coefficients of semiconductors for diffusion gratings are on the order of 0.5 to l.Ocrn-'. Equation (17) can be solved to give an expression for the beam intensities at the exit of the crystal:

I, = I , (B + WrLe - a L

B + erL I , = I (B + WrLe - o L 1

where L is the interaction length (the smaller of the overlap of the two beams or the physical crystal length), /?= I J I , is the ratio of the pump-tosignal-beam intensities, and I,, is the intensity of the signal beam at the entrance face of the crystal. A useful technique for measuring the gain coefficient in the laboratory is to measure the intensity of the signal beam with and without a strong pump beam (fi >> 1). The gain in the signal beam is given by I,(pump beam on) Y = I,(pump beam 00= p Thus the signal beam experiences an exponential gain with respect to interaction length. The gain coefficient can be readily calculated from Eq. (20); however, care must be taken to measure the scattered pump light and subtract it from the apparent gain. In cases where the gain is a significant fraction of the beam ratio (0.lB > y > l), the gain coefficient can be calculated from r = - l1n ( yfi L fi+l-y

)

In cases where the measured gain turns out to be comparable with the beam

330

JAMFS

E. MILLERD, MEHRDADZlARI AND

&SHIN

PARTOVl

ratio ( y z @), there is significant pump depletion, and the small-signal analysis is not valid. Equation (18) shows that the gain coefficient is proportional to the 90-degree spatially shifted component of the space-charge field divided by the modulation index. This can be written as

Physically, Em represents the space-charge field that occurs when m = 1, which is the maximum that can build up. Although the actual magnitude of the space-charge field is proportional to the modulation index, the gain coefficient is independent. From Eqs. (8) and (9),

Em =

Without an applied field (Edc= 0), it can be seen from Eq. (9) that the phase of the space-charge field is at exactly 90 degrees, or 7r/2. Thus, under diffusion-dominated conditions (i.e., without applying an external field to the crystal), the photorefractive grating is optimized for unidirectional energy transfer. Because the dominant mechanism for carrier transport is diffusion, and due to the symmetric nature of the process, the charge distribution remains in phase with the intensity pattern. The 90-degree phase shift can be viewed as a consequence of the electric field being proportional to the integration of charge along the spatial direction (in Poisson’s equation, integral of cos is sin). Under diffusion-dominated conditions (i.e., Ed, = 0), the maximum space-charge field is equal to ED Eq Em = ED

(24)

Eq

Situations where Cp # 4 2 (i.e., E , # 0) will be considered in Section V.

2.

SPATIAL FREQUENCY DEPENDENCE

The buildup of space-charge field is strongly dependent on the period of the grating. Figure4 illustrates the full functional dependence of Em on

5

~OTOREFRACTIVITY IN

SEMICONDUCTORS

331

2 1.5

-< 8

Y

!z

E

WE

1

0.5

0 Grating Spacing (NA*) FIG.4. The normalized maximum spacscharge field Em as a function of grating period. Note that the gain coefficient r is linearly proportional to Em through Eq. (18).

grating period (i.e., Eq. 24). At small grating periods, E D >> E,, and

Therefore, the gain coefficient increases linearly with grating period. In this so-called trap-limited regime, the space-charge magnitude is bounded by the quantity of trap sites between each grating period. As the grating period increases, more traps are available, and the maximum space-charge field grows. Eventually, at large grating periods, the buildup is limited by the carrier diffusion length. In the limit of large grating periods, E, >> E D (see Eq. lo), and Eq. (24) reduces to

This is the so-called diffusion-limited regime where the gain coefficient decreases linearly with grating period. Notice that the maximum spacecharge field (and r) is independent of the trap concentration in this regime. Measurement of the gain coefficient at large grating spacing can be used to determine reff because no other free parameters exist in the model. Figure 4 shows how Em asymptotically approaches the 1/A relationship at large grating periods.

332

JAMES

E. MILLERD,MEHRDADZIARl AND AFSHINPARTOVI

In the intermediate regime between large and small grating periods, Eq. (24) can be used to calculate the gain coefficient. The function is peaked at a grating period equal to

with a value of

Clearly, the larger the trap density, the higher is the photorefractive gain. By measuring the gain coefficient as a function of grating spacing and fitting Eq. (24) to the data, both the effective electro-optic coefficient and the trap density of the material can be determined. An example of this fit for a ZnTe crystal will be shown in Fig. 7.

3. INTENSITY DEPENDENCE Under weak illumination, the thermal generation rate cannot be neglected from Eq. (3), and the gain coefficient will have an intensity dependence. The inlusion of thermal generation is found to reduce the modulation index of the grating according to

where I,, = j?/s is the so-called saturation intensity (Le., the irradiance at which photoconductivity equals thermal conductivity). The intensity dependence can then be accounted for by replacing m with m’in the expression for the gain coefficient so that

where To is the saturated gain coefficient measured at large intensity (i.e., I >> l,,). The saturation intensity can be found experimentally by measuring the gain coefficient as a function of intensity and fitting Eq. (30). Figure 5 shows the intensity dependence of the normalized gain coefficient (T/To) at

5

PHOTOREFRACTIVITY IN SEMICoNDUCToRs

333

Intensity (mWcm2) FIG. 5. Normalized beam-couplinggain coefficient (r/To) as a function of incident intensity in CdTe. (From Partovi et al., 1990.)

a grating spacing of 1.5 pm for a CdTe sample (Partovi et al., 1990). From the fit of Eq. (30), a saturation intensity of 110pW/cm2 was found. This small value, typical of semiconductors, results in a very high efficiency, since very little power is needed to overcome thermal conductivity. 4. TEMPORAL RFSPONSE The speed at which photorefractive gratings build up to their steady-state value is a function of the optical generation rate, the grating period, and the carrier transport properties in the crystal. For continuous-wave (CW) or long-pulsed lasers (rpulsc>> r,), the time constant for grating formation or decay can be expressed as (Nolte, 1995)

where

is the drift length

334

JAMESE. MILLERD,MEHRDADZIARI AND AFSHINPARTOVI

is the diffusion length, &

Td

=Q

is the dielectric relaxation rate Q=-

al1epr,A

hc

-k

Od

od is the dark conductivity due to thermal generation of carriers, a, is the absorption coefficient in the material due to the active trap site, and p is the

carrier mobility. In the simple one-carrier model, I( is equal to the electron mobility. Equation (31) shows that the photorefractive time constant is equal to the dielectric relaxation time plus two additional factors that become significant at small grating spacings and/or large applied fields. The increase in response time that accompanies large applied fields can be interpreted as a consequence of the requirement that more charge be generated to screen a larger field (thus requiring a longer charge integration time). When the optical generation rate is much higher than the thermal rate, the intensity dependence can be factored out, and the time constant can be written as

F,, represents the energy density necessary to fully saturate the grating and, in general, is a function of the grating period. If the grating spacing is large compared with the drift or diffusion length, the time constant is given by

Thus the buildup time varies inversely with the writing intensity. The grating decay time also can be calculated using Eq. (33); however, the correct illumination conditions must be used (e.g., I = I,,, if the grating is stored in the dark). The mobility-lifetime product pr, is an important parameter in determining the grating formation time. In general, it is desirable to have p7r as large as possible for fast grating formation, as seen in Eq. (33). Figure 6 shows the measured grating formation time constant in

5

PHCTTOREFRACTIVITY IN

SEMICONDUCTORS

335

100

10

3

4 5 6 78910 20 Intensity at sample,I (rnW/crr?)

30

FIG.6. Grating formation time constant in ZnTe measured as a function of combined intensity at the sample. L = 804 nm. (From Millerd et al., 1996.)

a ZnTe:V:Mn sample as a function of writing intensity. The inverse relationship with intensity is clearly evident, and a saturation fluence of 0.2 mJ/cm2 is used to fit the data with Eq. (32). Under short-pulse illumination ( T IT~~ ) such , ~ as~ that~ from ~ nanosecond or shorter pulsed lasers, charge generation occurs within the carrier lifetime, and the formation time is limited by carrier diffusion. The diffusion time can be estimated by

Tpr

= T,, X

A2e -

47r2k,Tp

(34)

In the short-pulse limit, only the mobility and grating period are factors in formation time. Carrier mobility in semiconductors is typically quite large. For instance, in CdTe the mobility can be as high 5000cm2/Vs, which results in a diffusion time of 5 ps for gratings on the order of 2 pm (Valley et al., 1989). Thus ultrafast photorefractive response is possible using pulsed lasers. Because of the high peak intensities in pulsed operation, free-carrier gratings can be significant and often larger than the refractive index gratings caused by the space-charge separation. In addition, the different mobility of electrons and holes will affect the temporal evolution of the gratings at short time scales as well as the saturated value (Jarasiunas et al., 1993).

336

JAMES E. MILLERD,MEHRDADZIARIAND AFSHINPARTOVI

5. ELECTRON-HOLE AND MULTIDEFECT INTERACTIONS The single-defect, single-carrier model cannot explain the effects observed in many semiconductors. In many cases, and particularly at low temperature, the role of multiple defects and optical generation of both holes and electrons must be considered. By including generation of both carrier types, it is found that each carrier type tries to build up a grating of opposite sign. Modeling the full dynamics of grating buildup requires a numerical solution of the rate equations; however, under a surprising number of conditions, the effects of bipolar photoconductivity can be included using the so-called electron-hole competition factor (Strohkendl et al., 1986; Picoli et al., 1989). The effects of electron-hole competition can be included by modifying the gain coefficient:

r = <(Iy-

(35)

where < ( I ) is the wavelength-dependent electron-hole competition factor. Under the assumption of a large grating period compared with a carrier diffusion length, &Ican ) be written as

where I, is the total intensity, nT. and pT are the density of available electrons and holes at their respective trap sites, en and e,, are the total emission rates, and s, and sp are the optical cross sections for the electron and hole, respectively. The total emission rate is the sum of optical and thermal processes, so en = s,l, e',", where ek" is the thermal emission rate of the electron. If one carrier type is the dominant photoconducting species, then, t = f 1. Since electron-hole competition reduces the effective gain coefficient compared with a single-carrier model, the effect is sometimes accounted for through a reduced reff.In many materials, including 1nP:Fe and GaAs, the role of multiple defect levels has been found to give a strong temperature dependence of the electron-hole competition factor that can cause the dominant photocarrier to change with temperature and thus change the direction of the wave-mixing gain (Rana et al., 1992). There also may be a strong wavelength dependence of the electron-hole competition factor because the cross section for electrons and holes may change independently. For example, in GaAs at room temperature, the photorefractive response for ,i < 1.06 pm is electron-dominated and for I > 1.15 pm is hole-dominated (Partovi et al., 1989). In samples where both electrons and holes are photoactive, the gain coefficient can have an additional dependence on the grating period through because of the different mobilities of the carriers (Strohkendl el a/., 1986).

+

<

5 PHOTOREFXA(STIVITY IN SEMICONDUCTORS

337

n

d

'a 0.5 -

@41=4.17 pm/V

0 v

NE = 1.4 x 10 14

,,I

.-"-3.f.

3 0.4Q)

.r(

30.3-

--..--.

f

,

*.-..-- -..

I

Q)

8 0.2-

0

"

2

4

6

8

Grating period (m) FIG. 7. Beam-couplinggain coefficient as a function of grating period for ZnTe at 633-nm wavelength. The dashed line is a theoretical fit to a single-carrier,single-traplevel model. (From Ziari et al., 1992.)

Figure 7 shows the measured dependence of the gain coefficient, which is linearly related to the space-charge field, as measured in a ZnTe sample (Ziari et al., 1992). From this measurement the effective trap density and effective electro-optic coefficient can be determined. Comparison with values of r41 measured using other means gives an indication of the electron-hole competition in the sample. In this case, is approximately 0.93, which indicates very little electron-hole competition. Combining this information with independent electro-optic modulation experiments to determine the crystallographic orientation of the sample, one can determine whether the dominant photocarriers are electrons or holes (Glass et al., 1985). The trap density is lower in this material than typical photorefractive semiconductors, which results in a somewhat lower maximum gain coefficient.

IV. Four-Wave Mixing In four-wave mixing, two strong pump beams are used to produce a phase conjugate of a weaker probe beam. The process is illustrated in Fig. 8. The two pump beams should be collimated and counterpropagating (i.e., beams 1 and 2 are phase conjugates of each other). Depending on the grating strength, spatial phase, and relative intensities of the beams, the phase-

338

JAMES E. MILLERD, MEHRDADZlARl A N D AFSHINPARTOVl X A

'

~-

I

>z

L

0

'3 FIG. 8.

Configuration and beam designations for four-wave mixing.

conjugate beam can be much stronger than the incident probe beam. Four-wave mixing is useful for wavefront correction, holographic interferometry, optical correlation, and measurement of material parameters.

1. DEGENERATE FOUR-WAVE MIXING

Degenerate four-wave mixing (DFWM) can be described using a similar coupled-wave approach as in beam coupling. For a transmission-type grating, the relevant equations are (Cronin-Golomb et a/., 1984) d -A, dz

d

d -A

dZ d

--Al 2

2

-A2= dZ

a

m'

= ---A,

m' a --yA,--A,

2

m'

2

---A,--A, 3 - 2

m'

(37)

a

2 a

- A --yA,--A, dz , - 2 2

where mf = 2m+m

I,

+I, +I, +I,

and

y=

i2nAne-'b Am' cos 8

(38)

Notice that under the conditions of I, = I, = 0, the equations reduce to the two-beam coupling case, where I, = I, and I, = I,. For four-wave mixing, the two pump beams are usually much stronger than the probe and phase

339

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS I

.....................................................................................

0.001 I 0.1

I

1

10

1 Pump beam ratio (r = \/I,)

FIG.9. Phase-conjugate reflectivity as a function of pumpbeam ratio r for several coupling strengths and a diffusion-typerecording (4 = 4 2 ) .

conjugate beams, I, I , >> I,, I,. Under the assumption of undepleted pumps ( I , << I, I , ) and negligible absorption (a = 0), the effective reflectivity experienced by the probe beam is given by (Cronin-Golomb et al., 1984)

;,[:

R = L -

sinh(yL/2) - [cosh(yL/2 (1/2) In r)

+

I’

(39)

where r = I J I 1 is the ratio of pump beams. For a pure diffusion recording, 4 = 742, and therefore, y = r/2. Figure 9 illustrates the relationship between pump-beam ratio and reflectivity for a diffusion-type recording. A broad maximum is evident that is a function of coupling strength. Figure 10 shows how the reflectivity and optimal pump-beam ratio vary with coupling strength for two values of grating phase shift (4 = 0, 71/2). Note that the reflectivity can exceed unity, which corresponds to a conjugate beam of

................................... 1 0

1

2

3

4

Coupling Strength (TL)

5

6

0

4 1

2

3

4

5

6

Coupling strength gL)

FIG. 10. Maximum DFWM reflectivity and optimal pump-beam ratio as a function of coupling strength for two values of grating phase (4 = 0,7r/2). Here r = 2171.

340

JAMES E. MILLERD,MEHRDADZIARIAND AFSHINPARTOVI

greater intensity than the probe beam. For the case of 4 = 0, the reflectivity becomes infinite as the coupling strength approaches 2n.This corresponds to a self-oscillation condition (finite output with no input). In this case, the optimal beam ratio remains at 1 for all values of coupling strength. For a diffusion recording (b = a/2, the optimal pump beam ratio increases with coupling strength, and there is no value of coupling at which self-oscillation occurs.

2. DIFFRACTION EFFICIENCY MEASUREMENTS For an incoherent reconstruction beam (i.e., I , is incoherent with the other beams by virtue of wavelength, temporal delay, polarization, etc.), the diffraction of the reconstruction beam is analogous to replay of an ordinary thick hologram (Yariv, 1978). The spatial phase of the grating is unimportant for the reading beam because the reading and writing of the grating can be viewed as separate processes. The reconstructed beam is a phaseconjugate replica of the signal beam; however, chromatic aberrations will be present if the optical wavelengths are different. The magnitude of the phase-conjugate signal can be expressed as a ratio of diffracted to undiffracted light (output diffraction efficiency) (Petrov et al., 1991a):

where r' = TIE,I/lm(E ,) is proportional to the phase-independent magnitude of the space-charge field. The intensity of the reading beam will reduce the modulation index and should be included in the denominator of Eq. (2). For optimal diffraction efficiency, m x 1 (i.e., writing beams are of equal intensity, and the increased photoconductivity due to the reconstruction beam is negligible). For the small coupling coefficients typical of photorefractive semiconductors,and for a diffusion recording (r'= r) the diffraction efficiency is

Absorption or photochromic gratings, which arise from the differences in optical cross section of full and empty traps, can be significant when measuring diffraction efficiency. The diffraction efficiency due to photochromic effects is (Bylsma et al., 1988)

tf, = sinh2 (2::8)'

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS

341

where Aa is the change in the absorption coefficient in response to illumination. In GaAs:Cr, the diffraction efficiency from the photochromic effect can exceed the photorefractive effect(Bylsma et al., 1988). The output diffraction efficiency in Eqs. (40), (41), and (42) is the ratio of diffracted to undiffracted light. The ratio of diffracted to incident light (input diffraction efficiency) can be found by multiplying by (1 - RXe-"), where R is combined reflectivity of the two crystal faces, to account for Fresnel reflections at the interfaces and linear absorption through the sample. Under short-pulse conditions, the diffraction efficiency builds up with increasing writing fluence (Fabre et al., 1988). The energy dependence of diffraction efficiency can be modeled as (Millerd et al., 1996)

where F = pulse energy/unit area and qma, is the saturated diffraction efficiency (ie., measured under high-intensity CW illumination). Figure 11 shows the measured diffraction efficiency in a 2nTe:V:Mn crystal as a function of writing fluence with a 6-11s pulsed laser. The measured saturation fluence agrees well with CW measurements, and the diffraction efficiency agrees well with Eq. (40), where the gain coefficient (r = 0.325 cm-') was measured in CW experiments.

1.2 1.0 -

0.6 0.4 0.0

0.2 0.0

I

I

I

1

I

I

I

0

1

2

3

4

5

6

7

Writing fluence (mJ/cm2) FIG. 11. Diffraction efficiency measured as a function of writing fluence: 6 4 s pulse, 1. = 765 nm. Fit using Eq. (43) with q,, = 0.012% and F,, = 0.23 mJ/cm2. (From Millerd er al., 1996.)

342

JAW

E.

MILLERD,MEHRDADZIARIAND

AFSHIN PARTOVI

3. SELF-PUMPED PHASE CONJUGATION An interesting adaptation of four-wave mixing is the self-pumped phaseconjugate geometry. I n this configuration, a single beam is incident on the crystal, the pump beams are derived from amplified scattered light and/or redirection of transmitted light, and a phase-conjugate replica of the original beam is produced. The photorefractive self-pumped phase-conjugate mirror (SPCM) was first demonstrated by White et al. (1982). A photorefractive crystal was placed inside a linear resonator, and a phase conjugate of the input beam was generated. Feinberg (1982), Cronin-Golomb (1983), Chang (1989, and others have since introduced alternative geometries. Under ideal conditions, phase-conjugate reflectivities can approach 100% (CroninGolomb et a/., 1984). SPCMs have been demonstrated using photorefractive semiconductors with measured reflectivities of 11% (Bylsma et al., 1989). Figure 12 shows the geometry for several phase-conjugate mirrors. The theory for self-pumped phase conjugation using photorefractive four-wave mixing has been developed by Cronin-Golomb et al. (1984). The reflectivity of the SPCM can be derived from a coupled-wave analysis where pump depletion is considered but absorption is neglected. Calculation of reflectivity for the mirror may be found by including the proper boundary conditions and requires solution of an equation that may have multiple roots (Cronin-Golumb et a/., 1982). Reflectivity of the ring mirror is plotted as a function of coupling strength for various values of external mirror reflectivity in Fig. 13. The threshold condition of the ring mirror is given by

t 44)

w A) RING

B) LINEAR

'3

'I

C)DFCM

FIG. 12. Geometry and beam designations for the ring and linear self-pumped phaseconjugate mirrors and the double-pumped phase-conjugate mirror (DPCM).

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS

343

Coupling strength, l Z FIG. 13. Theoretical reflectivity of the lossless ring SPCM as a function of coupling strength for several values of external mirror reflectivity M. (From Millerd et af., 1992a.)

where M = M , M , is the product of the intensity reflectivities of the two external mirrors. In the limit of perfect external mirrors, M = 1, the threshold coupling strength is 2. For the case of the linear mirror, the threshold condition is TL=InM

(45)

which has a threshold of zero for perfect external mirrors. The effect of absorption and nonlinear photorefractive response on the double-pumped phase-conjugate mirror (DPCM) and the self-pumped phase conjugate have been examined by Wolffer et al. (1989) and Millerd et al. (1992) and follow similar approaches. It is convenient to introduce a normalized gain coefficient rnorm defined by rnorm

=

r/a

(46)

where a and r are the loss and gain coefficients, respectively. SPCM performance may be calculated as a function of sample length and r,,,,. Numerical solutions to the coupled equations can be obtained in a straightforward manner in the case of the iing mirror because the boundary conditions at z = L are well defined. Results for the ring mirror are shown in Fig. 14 for different values of rnorm. It can be seen that for a given value of r,,,,, there is an optimal length of sample such that any further increase will reduce performance due to the overall increase in loss. It was found that under certain conditions, the effects of absorption may be approximated by lumping the absorption losses into the external mirror

JAMES E. MILLERD,MEHRDADZIARIAND AFSHINPARTOVI

344

FIG. 14. Effects of absorption on the ring SPCM retlectivity. The parameter q = l/r,,,,, the relative amount of absorption. M = 1. Points are the results using the approximation of Eq. (47). (From Millerd et a\., 1992.)

reflectivity: M = M,M2exp(-2aL)

(47)

This approximation is valid when the coupling strength is well above threshold and when the photorefractive gain is not sensitive to intensity. Physically, what occurs is that for large coupling strengths, the reflectivity is limited by the loss and becomes equal to the effective transmission calculated using Eq. (47). Comparisons with the numerical solutions show that this approximation is very good at all coupling strenghs for rnorm > 10 (see data points in Fig. 14). For smaller rnom values, this approximation begins to underestimate reflectivity at coupling strengths around threshold. Investigation of the coupling strength necessary for threshold in the ring mirror reveals that for rnorm < 2, there is no length of sample for which oscillation may be achieved. This agrees with the results of Wolffer et af. (1989) for the double-pumped phase conjugator. The similarities between the two configurations have been identified previously by Cronin-Golomb (1990). Unlike laser resonators, simply having net gain r - a > 0 is not enough to guarantee oscillation. Because of this requirement, rnorm provides a good photorefractive figure of merit.

4.

POLARIZATION SWITCHING

With the proper choice of crystal geometry, cross-polarization coupling is possible in semiconductors (Yeh, 1987; Partovi et al., 1987). This type of

5 PHOTOREFRACTIVITY I N SEMICONDUCTORS

345

coupling, where the diffracted light is polarized orthogonal with respect to the polarization of the incident or zero-order ‘beams, allows for background suppression and high signal to noise. In a 43m crystal, the change in dielectric psmittivity caused by an electric field is given by

(48)

where x, y, and z represent the crystal axis, r41 is the electro-optic coefficient, and n is the bulk refractive index of the material. For two- and four-wave mixing, the coupling constant can be shown to be proportional to (Yeh, 1987) 1

Ti,ja- (ilAelj) n

(49)

where i and j are the polarization unit vectors of the diffracted and reading beams, respectively. For a grating (and electric field) long an arbitrary crystal orientation, Eqs. (48) and (49) can be used to determine the polarization of a diffracted wave from the grating. For example, using the geometry of configuration A in Fig.3 but rotating the crystal 90 degrees about the (110) axis results in a configuration where conventional parallelbeam coupling is prohibited. The grating and electric field are along the (1 10) axis (equivalent to p polarization) so that

The writing beams are polarized along the (001) axis (s polarization), and the conventional coupling coefficient is

However, the cross-polarization coupling coefficient (i.e., reading beam is s-polarized and the diffracted beam is p-polarized) is nonzero:

346

JAPES E. MILLERD,MEHRDADZIARIAND AFSHINPARTOVI

Thus it is possible to couple energy from one polarization state to another. Polarization coupling is very useful in optical signal processing because a polarizer can be used to isolate a weak diffracted signal from the writing beams (Cheng et al., 1987).

V. Enhanced Wave-Mixing Techniques Although semiconductors have the advantages of fast response times, sensitivity in the near-infrared, relatively low cost, and high optical quality, the magnitude of the photorefractive effect (e.g., measured through diffraction efficiency) remains small for diffusion-type recordings. That is, the maximum refractive index change, produced in response to holographic exposure and carrier diffusion, is relatively small. Even though photorefractive semiconductors have good sensitivity (index change per absorbed photon), the small saturated refractive index change results in relatively low diffraction efficiency and wave-mixing gains. In order to compensate for the small electro-optic coefficients and to achieve larger gain coeficients, researchers have investigated many methods of increasing the magnitude of the photorefractive space-charge field. This section will cover various gain-enhancement techniques, including the application of an external dc electric field (Albanese et al., 1986; Liu et al., 1988) the moving-grating technique (Kumar et al., 1987; Imbert et al., 1988), application of an ac electric field (Kumar et al., 1987; Klein ef al., 1988) and use of a temperature-dependent resonance (Picoli et al., 1989; Millerd et al., 1990). Gain coefficients as high as 16.3 and 31 cm-' have been achieved in GaAs and InP, respectively, which represents a significant improvement over the diffusion case (gain coefficients F = 0.6cm-') (Partovi et a/., 1990). The issue of photorefractive response at large modulation index (or large signal effects) in materials with applied electric fields will be summarized. Large signal effects are significant in all the gain-enhancement techniques presented here and must be accounted for when modeling device performance.

1.

DC APPLIEDFIELDS

The application of a simple dc field will increase the magnitude of the space charge field, as can be seen from Eq. (8). It will, however, also reduce the phase shift from 4 2 , as seen from Eq. (9). The reason for the enhancement in magnitude is that the carrier movement is now assisted by drift, so charge may be effectively moved at larger grating spacings.

5

347

PHOTOREFRACTIVITY IN SEMICONDUCTORS

Unfortunately, the symmetry is now broken, so the carriers move preferentially in one direction, causing the index grating to shift in phase. Figure 15 illustrates what happens to the space-charge field magnitude and phase as a dc field is applied. For this calculation it was assumed that E , = O.lE,; i.e., the grating spacing was large enough that diffusion played a small role, except for very small applied fields. Although the imaginary component (spatially shifted by 90 degrees) continues to increase for all values of electric field, the phase of the grating is not 90 degrees. At large grating spacings such that ED << Ed, << E4, the diffraction efficiency of the grating is given by 2

(53) Thus the diffraction efficiency increases as the square of the applied field. Figure 16 shows measured diffraction efficiency in a CdMnTe:V sample as a function of applied dc electric field; the quadratic dependence on applied field is clearly evident (Ziari et al., 1994). For beam coupling and self-pumped phase conjugation, maximum performance is obtained when the grating is shifted by 90 degrees. Several of the methods that have been employed successfully to shift the grating back to the optimal phase are discussed in the following subsections.

g

1

90 80

0.8

0

g. a

3

70

fu 0.4

6o 50

3 0.6

0a 0.2

40

Q

v)

0

0

0.5

1

1.5

2

30

Applied field @,/Ed FIG. 15. Dependence of space-charge field magnitude and phase on externally applied electric field strength. Plots are calculated from Eqs. (8) and (9), where E D is taken to be 0.1.5,. Field quantities are normalized by E,, the trap-limited field. Im(E,) is the imaginary component of the space charge. (From Nolte, 1995.)

348

JAMES E. MILLERD.MEHRDADZWRl

0

1

2

AND A S H I N

3

PARTOVl

4

Electric field kVlcm FIG. 16. Measured diffraction efficiency for a CdMnTe:V crystal as a function of applied dc electric field. Writing wavelength = 750nm, reading wavelength = 633 nm, A = lOpm, I>= 5mm.

2. AC FIELDS The asymmetry caused by the dc field and subsequent shift in grating phase can be avoided by periodically alternating the direction of the electric field. If the field is cycled much faster than the grating buildup time rPrbut slower than the carrier lifetime T,, then each carrier sees a constant field over its lifetime, but there is no net drift of the grating over time. If the period of t h e square-wave field is T,,, then (Stepanov and Petrov, 1985)

for the ac field technique to be effective. As Kc approaches zPr the grating begins to move with the field, and part of the electrical signal will be transferred to the optical beams. At the other limit, that is, T,, < rr, a single carrier experiences no net drift, since it sees fields in both directions over its lifetime. Violation of this condition will lead to a reduction in enhancement, but no signature of the ac signal will be imposed on the beams. Since E,, is proportional to the applied field squared (see Eq. S), the coupling direction is not affected by the sign of the applied field. It is, however, sensitive to the magnitude, so it is important to use square-wave fields with fast switching times (rise time qise << T,) in order to avoid having a signature of the ac field transferred to the optical beams during the finite switching time. The ac field technique has the decided advantage that no resonance needs

5

PHCXOREFRAC~IVITY IN SEMICONDUCTORS

349

to be maintained because the enhancement is insensitive to intensity, grating spacing, and applied field magnitude. The grating formation time is related to intensity, so as intensity is increased, care must be taken to ensure that the frequency of the square wave is still sufficiently high to meet the criteria of Eq. (54). The most significant problem using ac field enhancement is the electronic switching of large voltages at high speeds. The complexity of the switching circuit and the unwanted electrical interference that accompanies high-frequency alternating currents could be a prohibitive factor when using this technique in practical applications. The theory for square-wave enhancement is described by Stepanov et al. (1985). The first-order sinusoidal space-charge field is given by

where kg = 2?r/h,LEa = pzE,, is the carrier drift length, LD= (Dt)'/' is the carrier diffusion length, L, = (~kbT/e2Nt)'/'is the screening length, L, = EEac/eN,,u, is the carrier mobility, z is the carrier lifetime, E , is the magnitude of the applied ac field, and D is the carrier diffusion coefficient. Walsh et al. (1990) have shown analytically that the measured gain coefficient using a slew-rate-limited ac waveform can be significantly less than that predicted for a perfect square wave. In semiconductor materials, where the grating formation time is small, even at modest intensities, it is necessary to produce high-frequency square waves in order to fulfill the condition given by Eq. (54). As the driving frequency increases, it becomes difficult to maintain sharp square waves, because capacitive losses limit the slew rate. Ziari et al. (1992) have shown the dramatic effects of slew-rate limiting in CdTe and also have confirmed the prediction of Klein et al. (1988) who speculated that another phenomenon, field shielding, also would impose reductions on the measured gain. Field shielding arises when the ac period is comparable with or longer than the dielectric relaxation time and large spatial nonuniformities in the illumination exist across the width of the crystal. Charged carriers can build up in the darker regions and produce a field that opposes the applied one. This can be overcome by ensuring uniform reference-beam illumination (which may not always be possible) or by reducing the ac period so that T, << zd, where z, is the dielectric relaxation time, In semiconductors, dielectric relaxation times of less than 1 ps are typical for intensities on the order of 100 mW/cmz, requiring megahertz frequencies to overcome shielding. Field shielding also will be a problem in dc field enhancements where illumination is nonuniform, particularly because of electrode nonlinearities

350

JAMES E. MILLERD,MEHRDADZIARIAND AFSHIN PARTOW

0

5

10

I5

20

2

Electric Field (kV/cm) FIG. 17. Gain coefficient versus applied field for CdTe:V using ac field enhancement, 230 kHz. A = 7.5pm, I , = 75mW/cm2. Slew-rate rise time was 0.7 p. The simulation curves for perfect-square-wave, slew-rate-limited, field shielding are compared with experimental data at a beam ratio of lo4. (From Ziari er al., 1992.)

.f

=

(Ziari and Steier, 1993). Figure 17 shows measured and predicted performance of a CdTe:V sample, where both field shielding and slew-rate limiting are taken into account. Though both effects were important, slew-rate limiting was clearly the most severe limitation under these experimental conditions (Ziari et af., 1992).

3.

MOVING GRATINGS

By introducing a small frequency shift in one of the two beams, the interference pattern will no longer be stationary in time but rather will move in one direction or the other depending on the sign of the frequency shift. If the speed and direction of the interference pattern are matched carefully, it can compensate for the change in grating phase and restore the 7112 phase shift. In order to describe the situation mathematically, it is necessary to consider both the time and spatial dependence of the space-charge field. By introducing a frequency shift 6w in one of the beams, the grating will have a velocity u = Sw/k,. Allowing the quantities N,, n, J and E,, to be time-dependent in the material equations permits finding an optimum velocity for the grating velocity (Refregier et al., 1985) eN,sloA2

~ o p c=

- ~ 4 E ,7 ~ ~

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS

35 1

In order to arrive at this solution, it is necessary to assume that (1) the carrier drift length is larger than the grating spacing, pnrEd, > A, (2) the applied field is much less than the trap-limited field, E , << Eq, and (3) the diffusion field is negligible, ED << E,. These requirements are typically satisfied, since conditions outside these limits represent regions of small enhancement. It is interesting to note that in the limit of very small applied fields, Ed, << E D , the optimal velocity becomes

The fact that the velocity is linear in applied field illustrates that for small applied fields, the net drift of charged carriers may be compensated by moving the grating the appropriate amount. At the optimal fringe velocity, the space-charge field is given by (Imbert et al., 1988)

=m

where Ediff= l/(k#p,) is the equivalent diffusion field. At the optimal fringe velocity and grating period, the space-charge field is given by (Refregier et al., 1985)

The grating is moved at the rate of approximately one grating period per grating buildup time. In this way, the grating always has a 71/2 phase shift with respect to the intensity pattern and can efficiently transfer energy between the beams. To achieve a moving grating, one of the beams in the beam-coupling experiment must be shifted in frequency. Several techniques are available for this (Hu, 1983; Kothiyal and Delisle, 1984), including Doppler shifting of the pump beam by reflecting it off a piezoelectrically driven mirror (Imbert et al., 1988; Refreiger et al., 1985). Gain coefficients as high as 8 cm- have been measured in GaAs using moving gratings (Imbert et al., 1988).

’

352

JAMS

E. MILLERD,MEHRDADZlARl A N D AFSHINPARTOVl

For practical applications, the moving-grating technique is inconvenient because of the need to introduce a frequency shift that should change with applied field, intensity, and grating spacing, as shown in Eq. (57). In addition, some important applications of photorefractive materials such as self-pumped phase conjugation do not allow frequency shifts of individual beams because all the pump beams are derived from a single input beam.

4. TEMPERATURE-INTENSITY RESONANCE

Another mechanism similar to moving gratings, for obtaining the desired n,/2 phase shift, is the so-called temperature-intensityresonance (Picoli et a/., 1989). The resonance requires that thermal conductivity and photoconductivity be dominated by different carrier types. Because of this rather unusual requirement, the effect has only been reported in InPFe; however, it should occur in certain wavelength regions of GaAs and may be possible in other materials with appropriate deep levels. In order to describe the physics of the resonance, it is necessary to include both electron and hole optical and thermal generation and use a one-defect, two-band model. The full expression for the space-charge field has been derived by Picoli et al. (1989) and is given by

where

The 90-degree component of the space-charge field is found to be a maximum (i.e., E l is purely imaginary) when

where en is the total emission rate for electrons, ep is the total emission rate for holes, nT is the conception of traps containing electrons (Fez+ in the case of InP), and pT is the concentration of traps with holes available (Fe3+). Notice that if the thermal emission rates are not taken into account (that is, e;'' = 8," = 0), the fulfillment of the resonance condition in Eq. (62) also means that the electron-hole competition factor 5 in Eq. (36) is zero, resulting in a zero gain coefficient. If, as in the case of InP:Fe, thermal and

5 F'HOTOREFRACTIVITY

IN

SEMICONDUCT~RS

353

optical processes are dominated by different carrier types, then { will be nonzero. For most InPFe, the thermal emission rate of holes and the photoionization rate of electrons near room temperature can be neglected, so the resonance condition becomes

where ef,'l is the thermal emission rate of electrons, cP is the optical cross section for holes, and I , is the total intensity. The left side of Eq. (63) is a function of temperature alone, and the right side is a function of intensity, so it is possible to fulfill this condition by adjusting temperature and intensity accordingly. Under these conditions, the factor becomes equal to 112. Combining Eq. (60) and Eq. (18), the gain coefficient can be calculated as a function of temperature and intensity. Figure 18(a) plots the gain coefficient as a function of pump intensity for three different sample temperatures with trap densities and thermal and optical cross sections typical of InPFe. In practice, the resonance peaks are not as sharply peaked, due to absorption along the interaction length of the material. Figure 18(b) shows a typical plot of the gain measured as a function of intensity. The measured

<

10

0.

FIG. 18. (a) Theoretical curves for the gain coefficient r versus pump intensity at 1.06ym for three values of sample temperature. Equation (15) was used with A = l o w , E, = lOkV/ cm. (b) Fit of experimentally measured gain coefficient versus pump intensity in an 1nP:Fe sample ( T = 290 K, A = 6pm, Ea = 8 kV/cm) fit to a theoretical curve, calculated by integrating r along the thickness of the sample. (From Picoli et al., 1989.)

354

JAMES E. MILLERD.MEHRDADZIARIAND AFSHINPARTOVI

gain coefficient is really an average along the length of the crystal. Because of absorption and the resonance of gain with intensity, the peak gain coefficient occurs somewhere inside the crystal and may be higher than the measured gain coefficient. Indeed, it has been found that the largest gain coefficients are measured using a thin sample.

5. NEAR-BAND-EDGE EFFECTS The application of external electric fields serves to increase the overall magnitude of the space-charge field, while enhancement techniques such as moving grating, temperature-intensity resonance, and ac fields optimize the 4 2 spatial component of the space-charge field. Most of the work concerned with increasing the photorefractive effect in semiconductors, as well as oxide materials, has used these space-charge enhancement techniques along with the conventional Pockels electro-optic effect. The band-edge enhancement increases the magnitude of the index gratings, and hence the gain coefficient, through the use of nonlinearities that are present in semiconductors, in addition to the Pockels effect (Partovi et al., 1990). It is well-known that the application of an electric field to a material will change its absorption coefficient and refractive index near its band edge (Franz et al., 1958). This effect, known as the Franz-Keldysh efect, occurs because in the presence of an electric field E, the conduction and valence bands of a material are spatially tilted, thereby increasing the probability of photon-assisted tunneling across the bandgap (see Chap. 2). The changes in absorption coefficient (electroabsorption) and refractive index (electrorefraction) have been measured in InP and GaAs (Van Eck et al., 1986), Si (Soref and Bennett, 1987), and quantum well structures (Miller et a/., 1984). Electroabsorption and electrorefraction are related through the KramersKronig relationship. Although electroabsorption (EA) is stong near the band edge, it decreases rapidly with increasing wavelength for space-charge fields typical of the photorefractive effect ( < 100kV/cm). The peak of the electrorefraction (ER) spectrum, however, occurs at longer wavelengths and decreases more slowly with increasing wavelength, so ER is significant at wavelengths where the background absorption is relatively small. It is therefore possible to form strong electrorefractive gratings in spectral regimes with small background absorption. The magnitude of the index change due to ER can be many times that of the linear or quadratic Pockels effect for moderate electric fields (- 20 kV/cm). Using the space-charge field generated in photorefractive semiconductors through the usual drift and diffusion processes, it is possible to write gratings using the large near-band-edge electrorefraction nonlinearities. This method

5

f%OTOREFRACTlVlTY IN SEMICONDUCTORS

355

has the additional advantage that in the proper geometry, the conventional electro-optic grating also can add to the electrorefractive grating to result in even larger nonlinearities. The process of photorefractivity at wavelengths near the band edge can be divided into a discussion of band-edge electrorefraction photorefractivity (ERPR) and its combination with the conventional Pockels electro-optic photorefractivity (EOPR). Below the bandgap, the Franz-Keldysh electrorefractive change in the refractive index of a semiconductor due to application of a field can be approximated by a quadratic equation, AnERcc E2 (Alping and Coldren, 1987). This is different from the Pockels electro-optic effect, where An,, cc E. The sign of the refractive index change due to band-edge electrorefraction (ER) does not depend on the direction of the electric field. To first order, the ER effect is also independent of the polarization of the incident light. Under diffusion-only conditions (no applied field), for a sinusoidal photorefractive space-charge field, the induced refractive index pattern will, therefore, have twice the spatial frequency of the space-charge field. Since such a grating is not Bragg-matched in the direction of the writing beams, it is not possible to transfer energy in two-beam coupling with such a field. This problem can be overcome by application of an external field. An external dc electric field adds to the spatially varying field (see Eq. 7). Figure 19 shows the space-charge field and AnER for two directions of applied field and for no applied field. Without applied field, the index grating has twice the spatial frequency of the space-charge field pattern and cannot transfer energy. It can, however, be read via a third beam at the

Spacesharge field

Rehctivc index change

field 0

FIG. 19. ERPR grating formation.Without applied field, the grating period is doubled. The polarity of applied field controls coupling direction through the phase of index grating. (From Nolte, 1995.)

356

JAMES E. MILLERD,MEHRDADZIARIAM) AFSHIN PARTOVI

appropriate Bragg angle in a four-wave mixing geometry. Application of a bias electric field lifts the degeneracy, and the refractive index grating has the same periodicity as the space-charge field (Partovi and Garmire, 1991). The two directions of the applied electric field change the phase of the ERPR refractive index grating 180 degrees with respect to the space-charge field. If the phase of the space-charge field is shifted with respect to the intensity pattern, through a combination of diffusion, drift, and external techniques such as moving grating, the ERPR grating can couple energy. Because of the phase dependence of the refractive index on applied electric field, the direction of energy transfer is dependent on the field direction.This is in contrast to the conventional photorefractiveeffect, where application of a field can only increase the magnitude of the energy transfer without affecting its direction. The direction of EOPR energy transfer is dictated only by the crystal orientation and the dominant photocarrier species (Glass et al., 1985). Figure 20 shows the change in absorption coefficient due to applied electric fields from 1 to 12 kV/cm in a GaAs sample with indium-alloyed contacts (Partovi et al., 1990).The background (no applied field) absorption spectrum for this sample is also shown. It can be seen that large Aa values occur at a spectral region where the absorption coefficient is rapidly increasing with decreasing wavelength. The change in refractive index due to a change in the absorption coefficient can be determined by using the Kramers-Kronig relation:

lo

ch An(hv) = ; P

Aa(hv’)

(hV’)*

- (hV)*

d(hv‘)

(64)

Q $ 2

0 900

910

920 930 940

950 960

WAVELENGTH (nm) FIG.20. Franz-Keldysh electroabsorptivechange in the absorptioncoefficient as a function of applied field and wavelength. The zero-voltage absorption curve is also shown. The electroabsorption curves are for fields of E = I to 12 kV/cm in 2 k V / m increments, with larger values corresponding to larger fields. (From Partovi and Garmire, 1991.)

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS

357

where v is the optical frequency at which An is calculated, h is Planck's constant, Aa is the change in the absorption coefficient measured at frequency v', and P is the Cauchy principal value of the integral

A set of discrete Aai measurements can be made continuous through use of linear interpolation between each pair of hvi, hv,,, points. The change in absorption coefficient above the bandgap can be inferred by fitting the data to a broadened electroabsorption theory so that the index change in the presence of a field can be calculated (Partovi and Garmire, 1991). The result of applying the Kramers-Kronig relation to the electroabsorption spectrum is shown in Fig. 21. The absorption spectrum of the sample is also shown. For photorefractive applications, large An and small a values are desirable. We can see from Fig. 21 that at 1 > 905 nm, these requirements are simultaneously satisfied. However, if the change in the absorption coefficient due to electroabsorption is included, then the optimal wavelength region appears to be 1> -910nm. The change in absorption and refractive index near the band edge in response to electric field can be written as

5

1

1

1

1

1

1

1

1

1

1

4 9-

3

$*

4

1 0

-1 905

915

925

935

945

955

-.

WAVELENGTH (am) FIG.21. Franz-Keldysh electrorefraction as a function of wavelength and applied electric field in undoped GaAs calculated by applying the Kramers-Kronig relationship to data of Fig. 20. The zero-voltage absorption spectrum is also shown. (From Partovi and Garmire, 1991.)

JAMES E. MILLERD,MEHRDADZIARIAND AFSHINPARTOVI

358

where n,(A) and n,(A) and a,(A) and a,(A) are the linear (non-Pockels) and quadratic components of An@, E ) and Aa(R, E), respectively. Although electroabsorption can be used to write an absorptive grating, such a grating would exist in a spectral region with high background absorption and is not desirable for most photorefractive applications. Figure 22 is a comparison of the electrorefractive index change An,& = 905 nm) and the Pockels electro-optic index change An,, using no = 3.55 and r4, = 1.5 x 10-'2V/cm. It can be seen that at fields of E > 13kV/cm, the ER effect becomes larger than the EO effect due to its quadratic electric field dependence. It is therefore at large space-charge electric field values (such as achieved with moving grating or temperature stabilization) that using the electrorefractive effect provides dramatic improvements in beam-coupling gain coefficient. By substituting Eq. (7) into Eq. (64) and solving for the component of the refractive index that has the same spatial frequency as the grating, it is found that the gain coefficient is given by (from Eq. 18)

-

where

and it is assumed that

> E l . Unlike the linear electro-optic effect, there

Ed,

25 .I

20 h

n

f

U

6 Q

15

-

ELECTROREPRACTIVE

-

10

5 0

0

5

10

15

20

25

30

ELECTRIC FIELD (kV/cm) FIG. 22. Comparison of electrorefractive and electro-optic change in the refractive index as a function of applied electric field at L = 905 nm wavelength. (From Partovi and Garmire, 1991.)

5

PHOTOREFRACTIVITY IN SEMICONDUCTORS

359

is no crystal orientation dependence on the ERPR gain coefficient. Therecan be measured by selecting a crystal orientation and beam fore, rER polarization where beam coupling from the conventional electro-optic effect is not allowed. Likewise, it is possible to choose the crystal geometry, beam polarization, and direction of applied field so that the conventional EO grating adds to the ER grating for even larger beam-coupling gains. Furthermore, techniques such as moving grating and temperature stabilization that can be used to increase the 4 2 component of the space-charge field can be used to similarly enhance the ERPR or the combined ERPR and EOPR gain coefficients. Using moving gratings, a gain coefficient of r = 16.3cm- was obtained in GaAs (Iw = 140 m W/cm2, I,, = 29 pW/cmz, A = 7.0pm, L = 939 nm, E , = + 10 kV/cm) by optimizing the grating velocity. The 4-mm interaction length resulted in y = 600. The net gain achieved was r - a = 13.3cm-', resulting in more than 200 times net amplification of the signal beam (Partovi et al., 1990). Band-edge enhancement can be combined with the temperature-intensity resonance to obtain very large gains in 1nP:Fe (Millerd et al., 1990). The techniques are well suited for each other because they both require dc fields to operate. The combination offers a convenient method for improving gains, since it requires the use of only a thermoelectric device and a dc field. Figure 23 shows the measured gain coefficient as a function of incident intensity for two different sample lengths of InPFe at a fixed temperature.

Incident intensity mW/cm2 FIG.23. Gain coefficient versus incident intensity for two-wave mixing in two InP samples maintained at constant temperature: T = 2 0 T , applied field E, = 10 kV/cm, ratio of pump to signal intensity /3 = lo6, grating period A = 8 pm, 1, = 970nm, a = 5cm-'. (From Millerd et a/., 1992.)

360

JAW

E. MILLERD,MEHRDAD ZIARI AND AFSHIN PARTOVl

Large beam intensity ratios were used (B = 1.6 x lo6, the ratio of pump to signal intensity) to avoid pump depletion and large signal effects. The two samples were cut from neighboring positions in the same boule. The smaller gain coefficient measured in the longer sample arises primarily from the inability to maintain the optimal intensity over the entire length of the absorbing sample. Thus the measured gain coefficient is really a spatial average. Figure 24(a) shows the gain coefficient as a function of wavelength and applied field for a fixed grating spacing of 5 pm and a temperature of 19 "C. The pump intensity was adjusted to achieve maximum gain at each wavelength. Both absorption and gain increase with photon energy (decreasing wavelength), producing a maximum net gain in the region 970 to 985 nm. The effect of grating spacing on gain coefficient at a fixed intensity and temperature is shown in Fig. 24(b). The gain coefficient increases rapidly with grating spacing and shows a maximum near A = 9 pm. This particular optimum should change with the applied field. The direction of EOPR gain is determined only by the majority photocarrier species and the crystal orientation. The dominant photocarrier species in photorefractive media can thus be found by carrying out a single-beam electro-optic phase retardation experiment to determine the direction of the electric field for a positive index change and thus fixing the crystal orientation and then combining this result with the direction of energy transfer in a beam-coupling experiment (Glass et a!., 1985). By using this technique, other researchers have determined the majority photocarrier species at i,= 1064 nm to be electrons in undoped GaAs (Glass et a/., 1985)

FIG. 24. (a) Two-beam-coupling gain coefficient measured as a function of wavelength for InP sample. Intensity was adjusted at each point to achieve maximum gain for T = 19 'C, /j = 1ooO. Room-temperature absorption is shown as the heavy line. (b) Gain coefficient as a function of grating spacing for i. = 970nm. E. = 10 kV/cm. (From Millerd et 01.. 1990.)

5 PH~OREFRACT~VITY IN SEMICONDUCTORS

361

and holes in InP (Valley et al., 1988). Note that by using ERPR beam coupling, it is possible to determine the majority photocarriers by performing only a single experiment. This technique is therefore simpler and may prove useful for such determinations. 6. PHOTOREFRACTIVE RESPONSEAT HIGHMODULATION DEPTHS The application of external electric fields combined with the enhancement techniques have produced large two-wave mixing gains in semiconductor and sillenite materials; however, the largest gains are produced only for very weak signal beams (Imbert et al., 1988; Refregier et al., 1985; Stepanov and Suchava, 1987). Experimentally, it has been observed that as the signal beam intensity becomes comparable with the pump beam, a sharp decrease is measured in the effective gain coefficient, which cannot be solely explained by pump depletion (see Fig.25). The thin line in Fig.25 marked “pump depletion only” is the falloff in gain expected from standard pump depletion theory as the beam ratio approaches 1. The increased rate of falloff is due to the nonlinear relationship between the space-charge field and the modulation index. Note that p and m are related by m = (2fl”’)/(B + 1). This falloff is a significant concern for any application where a substantial fraction of the pump wave is coupled into the signal, such as self-pumped phase conjugation, in which the buildup of the phase-conjugate wave will reduce the effective gain and self-limit its performance. In fact, the effect of large signals must be taken into account when predicting performance of most any application where external fields are applied.

FIG.25. Two-wave mixing gain in 1nP:Fe measured as a function of input pump to signal beam ratio B. L = 4.0 mm, A = 7.7 pm, E, = 10 kV/cm, 1 = 970 nm, T = 20 “C. Circles are experimental values; bold line is fit to Eq. (69) with a = 4; thin line is standard pump depletion theory for 10 kV/cm case. (From Millerd et al., 1992.)

362

JAMES E.

MILLERD,MEHRDADZIARIAND AFSHIN P A R T ~ V I

Standard linear photorefractive theory, valid for a small modulation index, shows that the magnitude of the fundamental component of the space-charge field is linear with modulation index rn. The gain coefficient r IS therefore insensitive to modulation index, since r z E,/rn, where E , is the magnitude of the fundamental n/2 spatially shifted space-charge field. Evidence that the gain coefficient is a function of modulation index implies that the space-charge field must vary nonlinearly with rn. In the simplest physical description, this nonlinearity is caused by the clamping of the space-charge field magnitude to the applied field (Swinburne et al., 1989; Ochoa er al., 1986; Vachss and Hesselink, 1988; Au and Solymar, 1990). Several different approaches have been taken to model this nonlinear behavior. The finite-difference model developed by Brost (1992) accurately describes the performance of Bi,,TiO,, (BTO) in the presence of applied ac fields at all modulation depths and grating spacings. Because the model is not limited to steady state and calculates the full space-charge field (within the finite resolution of the element size), it allows calculation of the buildup of all harmonics. The model has led to a simple empirical formula that allows direct calculation of the large signal effects for materials using ac field enhancement in terms of the effective trap density, electro-optic coefficient, and carrier drift length. It also has been demonstrated experimentally that under certain conditions, the analytical model of Swinburne et al. (1989) can provide a reasonable estimate of the nonlinear behavior. Brost er al. (1992) generated a numerical model for the case of ac fields that assumes a single set of recombination centers and one species of free carrier. The functional form of the large signal effects can be described by an empirical equation E,(m)

=

where E,, was given by Eq. (22), and a/ is a fitting parameter. In addition, there is a simple relationship to calculate a / :

The thick lines in Fig. 25 are the fit using this empirical model, which has no free parameters. Equation (70) is a slightly modified form of the empirical function first suggested by Refregier er a/. (1985). Although determined empirically, this function allows direct calculation of the space-charge field at any grating spacing and modulation depth if N,, ref<,LD, and Ea are

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS

363

known. It should be noted that the model is not valid for all values of the applied field. Studies indicate that the expression is valid if the applied field is greater than twice the diffusion-limited field and less than the trap-limited field (ie., valid for 2E, c E, < E,). This does not impose a serious restriction, however, because the regions outside the boundaries represent regimes where enhancement from external fields is small and are therefore not of much practical interest. Typically, the conditions are satisfied for grating spacings larger than 1pm and applied fields larger than 2 kV/cm. 7. SUMMARYOF APPLIEDFIELD TECHNIQUES Each of the enhancement techniques has relative advantages and disadvantages. Table I summarizes the different space-charge enhancement techniques and their relative merits. The moving-grating technique is the least attractive from the standpoint that the optimal velocity is a function of intensity, applied field strength, and the grating spacing. Its strengths are that it can be used with any sample, regardless of carrier type, and can be combined with the band-edge enhancement. The temperature-intensity resonance condition can be fulfilled independent of applied field or grating spacing but does require that thermal conductivity and photoconductivity are dominated by different carrier types, a condition that cannot be achieved in all materials or at any given wavelength. The ac field technique is most attractive from the standpoint that no resonance must be maintained; therefore, it can be used over a broad range in intensity, applied field, and grating spacing. Unfortunately, the ac TABLE I COMPARISON OF ENHANCEMENT TECHNIQUE AND DEPMDENCE OF CONTROLLING PARAMETER ON INTENSITY, APPLIEDFIELD, AND GRATING SPACING ~~

Enhancement Technique Moving grating Temperatureintensity ac Field

~~

Controlling Parameter v, grating

velocity 7; temperature of sample T,,, waveform period

~

~~~~

Applied Field E

Grating Spacing

Can Use

A

ERPR

v aI

val/E,,

vah'

Yes

Toc -In(I)

None*

None*

Yes

T,
None*

None*

No

Intensity I

*The gain coefficient is a function of applied field and grating spacing; however, there are no special requirements to fulfill the resonance condition.

364

JAMES

E. MILLERD,MEHRDAD ZIARIAND AFSHINPARTOVI

field technique cannot be used with the band-edge enhancement because of the directional dependence of the gain coefficient on applied field. The bandedge enhancement technique (or ERPR) has characteristics that may prove useful in new devices (e.g., the direction of beam-coupling gain depends on external field direction). With the recent introduction of photorefractive semiconductor alloys, it is possible to band-gap engineer materials to take advantage of ERPR gains over a broad range of near-infrared wavelengths. For the application of semiconductors to holographic interferometry, where beam coupling is not necessary or perhaps not even desired, considerable enhancements in diffraction efficiency can be made simply by using dc fields and working near the band edge.

VI. Bulk Semiconductors The requirements that have traditionally been associated with photorefractive materials have been (1) presence of sufficient density of deep levels to create a space-charge field and (2) lack of a n inversion center of symmetry permitting a Pockels electro-optic effect. A high density of deep levels can be introduced through native defects and by doping with impurities. A great deal of knowledge already exists on the energy levels occupied by various impurities in many different semiconductor compounds (Pankove, 1971). The majority of photorefractive semiconductor investigations have been with the III-V materials GaAs and InP, mainly due to their ready availability in bulk semi-insulating form. CdTe and ZnTe appear to offer superior photorefractive properties and operate at more desirable wavelengths; however, they have only recently become available, and little experimental work has been done to grow high-quality II-VI materials. GaP has been explored as a visible (red) photorefractive material and has been shown to have a photorefractive response similar to the other 111-V materials. Advances in ternary and quaternary semiconductor growth techniques, both bulk and thin film, otl‘er the possibility for “band-gap engineered” materials that can be grown for maximum performance at most any desired wavelength. CdMnTe is an example of a recently explored ternary material that has demonstrated photorefractive response in the desirable 600-nm to 1-pm range. Growth of ultrathin (- 1 to 2pm) quantum well structures introduces an additional mechanism (layer thickness) to engineer the band gap of these materials. A convenient figure of merit for photorefractive materials is sensitivity: S=

rLR cos e 47clrJl - e-aL)

5

PHOTOREFRACTIVITY IN SEMlCONDUCToRS

365

where r p is the grating formation time, a is the absorption coefficient of the material, and L is the interaction length. By assuming L = 1cm, S would correspond to index change per absorbed energy per unit volume. Equation (71) assumes that the gain coefficient is independent of modulation index. For situations where applied fields are used, the gain coefficient measured at low modulation index (high beam ratios) will be larger than that measured at unity, thus overestimating the sensitivity of figure of merit for small beam ratios. Two other useful figures of merit are the normalized gain coefficient rnorm = T/a and the normalized electro-optic coefficient ~ n ~ r , , , , kmorn , . is a particularly important figure of merit for applications such as self-pumped phase conjugation, where gain must exceed loss by a factor of 2. The normalized electro-optic coefficient corresponds to the index change per separated charge and is a rough measure of efficiency. Normalization by the dielectric constant is necessary to compare measured data. 1. GaAs

Gallium arsenide can be made semi-insulatingand photorefractivethrough use of the stoichiometry-related EL2 center (Klein, 1984) or by doping with chromium (GaAsCr) (Glass et al., 1984). The growth technology for GaAs is arguably the most advanced of all the compound semiconductors; however, photorefractivequality can vary considerablybetween samples obtained from even the same boule. Photorefractive response has been measured from close to its band edge at 915 nm (Partovi et al., 1990) to 1.3pm (Cheng and Liu, 1990). Based on absorbance measurements, GaAs:Cr should have a photorefractive response as far into the infrared as 1.8 pm (Glass et al., 1984). Gain coefficients as large as 0.4 cm- have been demonstrated in diffusion-type recordings and as high as 16 cm- with the application of external electric fields and near-resonant effects (Partovi et al., 1990). Application of electric fields to GaAs results in spatial nonuniformities. It has been found experimentallythat for electric field values up to 2 kV/cm, high field domains exist across the sample that travel toward the positive electrode. These domains are attributed to negative differential resistivity arising from field-enhancedcapture of carriers by deep level traps (Rajbenbachet al., 1988). At applied fields larger than 2kV/cm, the fields are essentialy uniform. However, above 3 to 5 kV/cm, the mobility (and hence drift length) of the electrons is severely limited, due to intervalley scattering (Seeger, 1989). This limits the amount of space-charge increase that can be gained by applying electric fields. In practice, the highest applied fields that can be used in GaAs without causing field nonuniformities are about 4 kV/cm.

366

JAMES E.

MILLEXD,MEHRDADZIARIAND

AFSHINPARTOVI

2. InP Indium phosphide has been demonstrated to be photorefractive by doping with iron (InPFe) (Glass et al., 1984) and titanium (InPTi) (Nolte et al., 1989). Photorefractivity has been demonstrated from near the band edge at 970nm (Millerd et af., 1990) out to 1.32pm (Strait et al., 1990). Titanium-doped material is believed to have a photorefractive response past 2 pm, based on photoconductivity data. In addition, the bipolar conductivity found in some InPFe samples is not seen in InPTi material, mitigating electron-hole competition and optimizing photorefractive response. Net gain (r > a) has been demonstrated in 1nP:Fe at a wavelength of 1.32pm without using applied fields (Strait et al., 1990). The gain coefficient of 0.27cm-* was almost twice as large as the absorption coefficient, a = 0.145 cm-'. Using applied fields, gain coefficients as large as 30cm-' were measured at 980 nm, where the absorption coefficient was 5 cmThe role of multiple defects has been found to be important when modeling the photorefractive behavior of certain InP samples (Rana et al., 1992). A two-defect model has been shown to more accurately model the temperature-dependent photorefractive effect in InPFe rather than a single Fe defect model. The temperature-dependent concentration of these multiple trap levels leads to dramatic changes in electron-hole competition, which can result in sign reversal or quenching of the coupling coefficient. The concentration of this second defect appears to be enhanced near the seed end of the boule. However, in many cases, particularly for samples where photoconductivity is strongly dominated by holes, the single-trap model can give adequate results. Indeed, the single-trap model is adequate to describe the temperature-intensity balance that can take place in InP:Fe, resulting in large wave-mixing gains in the presence of dc applied fields. Field domains, similar to those measured in GaAs, also have been measured in InP; however, they are only present at low temperature (T < 77 K) (Nolte er al., 1990). The domains are a result of negative differential resistivity caused by a field-enhanced carrier capture into a deep level (not associated with Fe). Thermal ionization at room temperature depopulates this trap level, and field domains due to this mechanism are not present. A second mechanism for generating field domains has been reported at room temperature in InP, when used in a double-pumped phaseconjugate arrangement. In this case, field domains arise from negative differential resistivity caused by the sudden increase of intensity fringe contrast inside the crystal as the DPCM achieves threshold. The effect of these domains can be mitigated using a two-zone DPCM (Wolffer and Gravey, 1994).

'.

5

PHOTOREFRACTIVITY IN

367

SEMICONDUCTORS

3. GaP Growth of semi-insulating gallium phosphide (Gap) has been challenging; however, Kuroda et al. (1990) have reported photorefractivity in a nominally undoped sample. GaP has the highest bandgap of the 111-V semiconductors (2.26 eV, 550 nm), permitting operation into the visible. Beam-coupling coefficients of 0.33 cm- were reported at a wavelength of 633 nm. With applied electric fields and moving gratings, gain coefficients as high as 2.5 cm-' have been reported (Ma et al., 1991). Absorption measurements indicate that photorefractive response should extend out to 1 pm.

-

'

4. CdTe

Photorefractivity has been demonstrated in cadmium telluride by doping with vanadium (CdTe:V) (Partovi et al., 1990) and titanium (CdTe:Ti) (Bylsma et al., 1987). CdTe has by far the largest experimentally measured sensitivity figure of merit (index change per absorbed photon). Photorefractivity has been measured at wavelengths from 1.0pm to as long as 1.5 pm (Partovi et al., 1990; Schwartz et al., 1994). Photorefractivity should be possible at wavelengths even longer than this. In the absence of applied fields, gain coefficients as large as 0.7 cm- have been measured in CdTe at a wavelength of 1.32pm. Using an ac field for enhancement, gain coefficients as large as lOcm-', with a field of 10 kV/cm and a beam ratio B = lo4, have been demonstrated (Ziari et al., 1992). The growth of high-quality semi-insulating CdTe crystals has so far been difficult to reproduce. With improved growth techniques, CdTe could become an important photorefractive material.

'

5. ZnTe

Vanadium-doped zinc telluride (ZnTe:V) has been shown to be photorefractive in the wavelength region from 0.63 to 1.3 pm (Ziari et al., 1992). This material has the highest measured value of normalized electro-optic coefficient & ~ : r ~ (corresponding ~~/&~ to the electro-optic index change per separated charge) of the compound semiconductors shown in Table 11. This suggests that if high trap densities can be introduced, gain coefficients comparable with CdTe could be obtained. This material is sensitive in a spectral region compatible with a wide range of laser diodes. Beam coupling has been demonstrated using a GaAs-GaAIAs diode laser at 830 nm (Ziari et al., 1992).

TABLE I1 PERFORMANCE COMPARISON OF BULKP H O ~ R E F R A SEMICONDUCTORS CT~VE AND ENHANCEMENT TECHNIQUES Material* (enhancement) (ref.) GaAs:Cr (Klein, 1984) 1nP:Fe (Valley et a/., 1988)

1nP:Fe (Strait et a/., 1990) G a P (Kuroda et al., 1990) CdTe:V (Partovi e t a / . , 1990) Cd,5 Mn,,Te (Ziari et a/., 1994) ZnTe (Ziari et al., 1992) CdS (Taybati et a/., 1991) GaAs (MG) (Imbert et a/.. 1988) CdTe:V (AC) (Ziari et a/., 1992) 1nP:Fe (T-I) (Picoli et a/., 1989) GaAs:Cr (BE, MG) (Partovi et a/., 1990) I n P F e (BE, T-I) (Millerd er a/., 1992)

i , prn

OGrcrr. pm,N (norm)

3, em-'

Tmr..,cm-'

1.06 1.06 1.32 0.633 I .5

43 (3.3) 27 (2.23) 36.3 (3.0) 44 (3.7) 91 (8.9)

1.2 1.o 0.145 2.0 1.8

0.4 0.15 0.273 0.33 0.6

0.02 0.1

0.74

-

1.5

s( x 109,

I,, mW/cm*

rnorm = Pa

3 2.8

4Ooo 1070 2000 30 2.6

0.33 0.15 1.88 0.165 0.33

0.3

2

to0

0.2

11

32 48 37

T , , , ~ ~ , , ms

-

An, cm3/J

60 19

-

21 510

0.633 0.633 t .06

llO(1l) 40 (4)

5.2 0.5 1.65

0.45 0.3 7

0.0 I5 1.o 40

4700 80 50

0.086 0.6 4.2

I .32

91 (8.9)

2

10

10

75

5.0

162

6

-

10

3.0

-

-

140

4.65

-

100

120

6.3

21

-

1.06

-

2

0.939

-

3.5

16.3

0.97

-

5.1

32

Nore: Measured or calculated parameters were taken from referenoes as noted. Dash indicates parameters not reported. 'BE. band-edge enhancement;MG. moving grating; AC. ac field T-I, temperature intensity resonance; (norm), normalized by relative dielectric constant, 4.

5

369

PHOTOREFRACTIVITY IN SEMICONDUCTORS

'

The maximum gain coefficient measured at 633 nm was 0.45 cm- and fell to 0.12cm-' at 830nm. This sharp falloff in gain could be due to wavelength-dependent electron-hole competition, possibly due to multiple defect levels. Photoluminescence data also give evidence of multiple defect levels in this material. To date, however, no investigation of the temperature dependence of the photorefractive effect has been made to confirm this.

6. CdS Photorefractivity has been reported in semi-insulating cadmium sulfide (CdS) at a wavelength of 633nm (Taybati et al., 1991). The bandgap of this material, 2.47 eV, should allow operation into the green (- 520 nm). CdS belongs to the 6-mm symmetry group and as such is one of the few semiconductor materials not in the 43m group. It has three nonzero electro-optic coefficients, r13 = rZ3,r33 and r42 = r5', and is uniaxial, no = 2.46, ne = 2.48. Gain coefficients as high as 0.3cm-' were measured at 633 nm, where the background absorption was 0.5 cm- The grating formation time was measured to have a sublinear dependence on writing intensity due to the interaction of shallow traps in the material. The good sensitivity and normalized gain coefficient measured with diffusion fields make this a promising material.

'.

7. BULK11-VI ALLOYS Novel photorefractive materials have been developed by the growing bulk crystals of alloyed ternary 11-VI semiconductor compounds. Bandgap engineering of ternary alloys can serve photorefractive material technology by providing a tunable sensitivity window (Partovi et al., 1990; Ziari et al., 1991). One application of bandgap tunability would be to slightly blue shift the bandgap of CdTe such that the photorefractive-sensitive wavelength range can overlap with those of GaAs/AlGaAs semiconductor lasers. Another advantage of the alloying process is that certain composition of alloys offer a better overall material quality. The bandgap of the ternary alloys is a function of the mole fraction or the mixture ratio and, depending on the choice of the alloy, can be larger or smaller than that of the binary compound. For example, CdZnTe and CdMnTe both have bandgaps that are larger than CdTe, while alloying with HgTe results in lowering the bandgap energy. Figure 26 is an illustration of the bandgap energies of CdZnTe, CdMnTe, and HgCdTe as a function of mole fraction x (Brice and Capper, 1988; Willardson and Beer, 1982; Lay et al., 1986). The dielectric

370

JAMES E. MILLERD,MEHRDADZIARIAND AFSHINPARTOVI 3.5

80" K

MnTe

3.0 2.5

2.0

CdTe

1.5 1.o

0.5

I

I

I

I

I

FIG. 26. The bandgap energy of Cd, _,Zn,Te, Cd, -*Mn,Te, and HgxCd, -,Te ternary alloys as a function of mole fraction x.

and electro-optic properties of ternary alloys are expected to be near the average of the dielectric properties of the constituent binary compounds. The photorefractive properties of Cd, -,Zn,Te samples with x = 0.04 and .Y = 0.1 compositions have been studied (Ziari et al., 1991; Ziari, 1992). Figure 27 shows the grating-period dependence of the two-beam coupling gain at 1.064 pm wavelength. The fit to data suggests a trap density of 1.12 x iO'5cm-3 and a tiql product of 4.16pm/V, which are very close to the measured values in CdTe:V. Note that alloying has not decreased the
Cd,,Zn,,Te:V

r . . . " ' . " ' " " ~ ' ~ ' ' . . ~ . I . . . . l . . ~ '

0

I

2

3

4

5

Grating Period (pm) FIG. 27. Grating period dependence of two-beam coupling gain in Cd,.,Zno,, Te:V.

5

PHOTOREFRACTIVITY IN SEMICONDUCTORS

371

of dopant incorporation in ternary alloys can differ from the binary constituents. Sefler et al. (1992) have reported on photorefractivity in the bulk ternary material Cd,.,Mn,. ,Te grown nominally undoped, and recently, Cd,.,, Mn,.,,Te has been grown semi-insulating by doping with vanadium (Ziari et al., 1994). Photorefractivity has been demonstrated at wavelengths between 0.7 and 1.0 pm, which are compatible with laser diode technology. Because this is a dilute magnetic semiconductor, magneto-optic interactions such as the Faraday effect are present. Sefler et al. (1992) demonstrated that at low temperatures (to increase the paramagnetic susceptibility), a magnetic field can control the direction and magnitude of photorefractive energy transfer by rotating the polarization of the beams inside the crystal. Many other 11-VI semiconductors such as ZnS, ZnSe, and CdSe remain to be developed and investigated for photorefractive nonlinearities. Of the 11-VI materials, however, CdTe has the highest sensitivity figure of merit, suggesting that this material should continue to be explored. The wide possibilities of substitutional dopants, native growth-related defects, and alloying within the 11-VI material systems leave a vast potential for future research in photorefractive semiconductor materials growth and characterization. A comparison of bulk semiconductor photorefractive materials is provided in Table 11. The parameter values were taken from actual measurements, as noted by the references. Missing parameters are so noted. Three figures of merit were calculated, based on the published values: the northe normalized electro-optic coefficient malized gain coefficients rnorm, tn;reff/c,., and sensitivity S. The value for rmeas is the maximum gain coefficient measured in the specified reference. In some cases, the grating may have been measured with a lower gain, which formation time.,,,,z, results in a smaller sensitivity. Table I1 includes performance figures for both diffusion-type recordings and drift-type recordings where externally applied electric fields have been used to enhance the gain coefficient. Notice that the dramatically while the sensitivity use of these techniques increases rnorm figure of merit drops. Indeed, the increase in gain coefficient is offset by an increase in grating buildup time.

VII. Multiple Quantum Wells

One of the more interesting prospects for new photorefractive materials is multiple quantum well (MQW) structures. Thin layers of material are grown epitaxially on semiconductor substrates. The bandgap of the struc-

372

J A M E ~E. MILLERD,MEHRDADZIARI

AND h H I N

PARTOVI

ture is determined by the material alloy and the thickness of the quantum wells. Quantum confinement leads to strong exciton absorption that exhibits a quadratic electro-optic effect. The quantum wells can be made semiinsulating through an ion-implantation or deep-level doping process so that space-charge fields can be generated via trapping of charged carriers. Wavelengthsat or near the exciton absorption peak experience large changes in refractive index in response to the space-charge fields (Nolte et al., 1990). Because of the high absorption near the band edge and the difficulty in growing thick devices, the MQWs are typically only a few microns thick, and in most geometries, the substrate must be removed in an etching or polishing process. Alternatively, they can be grown on integrated Bragg mirrors. The thin gratings result in multiple diffracted beams; however, diffraction efficiencies as high as 3% in the first order have been reported in a 2-pm-thick sample containing GaAs/GaAlAs quantum wells (Partovi et al., 1993). This diffraction efficiency is comparable with bulk semiconductor structures thousands of times thicker. Two-beam coupling has been reported with gain coefficients as high as lOOOcm-' (Lahiri et al., 1996). Figure 28 shows two geometries for multiple quantum well photorefractive (MQW-PR)devices. Applied electric fields are required to assist in carrier transport and to obtain substantial diffraction efficiency due to the nature of the quadratic electro-optic effect. Figure 28(a) shows a geometry where the electric field is applied parallel to the MQW layers. This geometry is the easiest to produce because fabrication of contacts is necessary on only one side of the device. Charge transport is drift-dominated, and the formation of the space-chargefield is similar to that in bulk material. In this case, the Franz-Keldysh (F-K) effect, which is caused by electric field ionization of the exciton, results in a change in the absorption and refractive index near the bandgap of the material.

E

W

\E2/

cmaot. +

(a)

0)

FIG. 28. Photorefractive multiple quantum well device geometries: (a) parallel electric field; (b) perpendicular electric field.

5

PHCFTOREJXACT~VITY IN SEMICONDUCTORS

373

Figure 28(b) shows a perpendicular field geometry that is more difficult to fabricate but can have larger diffraction efficiencies due to the higher electric field strengths across the quantum wells. Application of an electric field perpendicular to the quantum well layers in a MQW results in a large shift in the exciton wavelength through the quantum confined Stark effect (QCSE) (Miller et al., 1985). A major advantage of the QCSE geometry is that since the voltage is applied in a direction perpendicular to the layers, which are typically 1 to 2 pm thick, very high electric fields (over 100kV/cm) can be obtained with small applied voltages (10 to 20V). At these high fields, the QCSE nonlinearities are much larger than the Franz-Keldysh effect (see Chap. 2).

1. MQW-PR DEVICESUSINGTHE QUANTUM CONFINED STARKEFFECT In a geometry with the electric field applied perpendicularly to the MQ W layers, absorption coefficient changes as large as 7500cm-' and a corresponding index change of An = 0.06 can be obtained with fields of 150 kV/ cm (Partovi et al., 1993). Since the field is applied perpendicular to these thin epitaxial layers (typically 1 to 2pm thick), such large fields are easily obtained by low voltages. MQW-PR devices in the QCSE geometry were first studied in 11-VI materials with excitonic peaks near 600 nm (Partovi et al., 1991, 1992). Figure 29 shows the exciton peak due to n = 1 heavy-holeelectron transition in a molecular beam epitaxy (MBE) grown sample 25000

8,.

575

, , I , ,

,

, I , ,

,,

I , ,

585 595 805 WAVELENGTH (nm)

, '1

615

FIG. 29. Absorption spectrum of a CdZnTe/ZnTe MQW before (solid line) and after (dashed line) ion implantation with hydrogen ions at a dose of 10L3/cm2at 600 keV. The excitonic feature is unaffected, while the resistivity is increased to 108Q-cm.

374

JAMES E. MILLERD,MEHRDADZ I ~AND I AFSHINPARTOVI

consisting of 90 periods of 51-A Cd,,,,Zn,.,,Te wells and 122-A ZnTe barriers grown on a buffer layer of 1.7pm of Cd,~,Zn,~,Te and 0.7pm of ZnTe on a semi-insulating (SI) GaAs substrate. The thick buffer layers are required in this material because of large lattice mismatch between the epitaxial MQW and the substrate material. In order to obtain semiinsulating behavior throughout the device, the MQW structure was ionimplanted with protons with a dose of 1013/cm2at 600 keV. The excitonic absorption is unaffected by ion implantation, while the resistivity is increased to greater than 1O8R-cm. Proton implantation is a common method used in epitaxial layers to reduce carrier lifetime. However, care must be taken not to broaden the excitonic absorption by excessive reduction of carrier lifetime with too high of an implantation flux (Silberberg et al., 1985). A photorefractive MQW device was fabricated from an ion-implanted section of the wafer by first evaporating 150nm of phosphosilicate glass (PSG) and a thin transparent electrode consisting of 20 8, of Ti followed by 80A of Au on the surface of the epitaxial layer. Next, a thick Au contact pad was evaporated on one corner of the device. After mounting the sample on a sapphire substrate, the GaAs substrate was removed by mechanical polishing and chemical etching. Dielectric and transparent electrode layers as well as an Au contact pad were similarly evaporated on the front side of the sample. In addition, during processing, each side of the device was ion-implanted through the PSG layers with oxygen ions to a dose of 10'3/cm2at 70 keV. The dose and energy of this implantation were designed to further increase the trap density and to further reduce the conductivity in a 50-nm-thick layer around the interface by an order of magnitude, thereby reducing lateral diffusion of carriers. After etching a hole through the sample to the contact pad on the backside of the sample, electrical contacts were made with gold wires. Wave mixing was carried out by intersecting two equal-intensity copolarized beams from a dye laser at d = 596nm to produce an interference pattern with A = 20pm, while an ac square voltage with -30 kHz frequency was applied. The interference pattern causes nonuniform charge excitation followed by transport to the MQW-dielectric interface. The screening of the applied electric field at the intensity maxima of the interference pattern reduces the magnitude of the QCSE at these locations. In this way, an absorption and refractive grating in phase with the intensity pattern is created in the device. Selfdiffraction of the two beams from the resulting Raman-Nath (thin) grating results in multiple-order diffraction at the output of the device. Output diffraction efficiency is defined as the ratio of the first-order diffraction beam intensity to the zeroeth-order transmitted beam. For a sinusoidal absorptive and refractive index grating with magni-

375

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS

tudes Aa’ and An’, the diffraction efficiency is given by

where J , is the first-order Bessel function, L is the sample thickness, and 0 is half the angle between the beams. Figure 30 shows the output diffraction efficiency qout as a function of applied voltage for this device. The corresponding ac voltage period is also shown. A maximum qOut= 1.35% and submicrosecond response was obtained at tpp, = 60 V (i.e., 150kV/cm). Availability of inexpensive, high-power, highly reliable laser diodes at 780 to 850 nm places particular importance on devices operating in this range. MQWs based on GaAs/AlGaAs material have excitonic features in this wavelength region. A great deal of research on this material system and electro-optic devices based on it has been carried out. Semi-insulating material can be obtained by ion implantation (Partovi et al., 1992; Silberberg et al., 1985; Glass er al., 1990) doping with a deep level such as Cr (Partovi et al., 1991), or growth at low temperature to result in excess arsenic incorporation, which depletes carriers in the material (Nolte et al., 1992, 1993).

8 1.5 c 2 1*25

45

I - ’ 0

0.75

UJ

0.5

F

0.25

15

u - 0

10

LL

@

0

0

10

20

30

40

50

80

70

VOLTAGE wolb)

FIG. 30. Diffraction efficiency as a function of applied square voltage amplitude at

I = 596 nm for beams of equal intensity with total incident intensity I = 3.6 W/cm’ [solid line). The frequency of the applied voltage was optimized at every voltage value for maximum diffraction efficiency. The values for the optimal voltage period are also shown (dots). (From Partovi et al., 1992.)

316

JAMES E. MILLERD,MEKRDADZIARIAND AFSHINPARTOVI

The first GaAs/AlGaAs PR-MQW devices in the QCSE geometry were made semi-insulating by Cr doping and consisted of 155 periods of Crdoped (1016cm3) 100-A GaAs wells and 35-A AI,,,Ga,.,,As barriers grown on top of a 1500-l$ AI,,,Ga,~,,As etch stop layer (Partovi et al., 1991). The top five periods were grown at low temperature (380"C), while the rest of the structure was grown at normal growth temperatures (630 "C). The low-temperature-grown (LTG) layers are incorporated to provide subpicosecond recombination times (Knox et al., 1991) to assist carrier trapping at the MQW-dielectric interface. Use of thin barriers (35A) improves the performance of the device by allowing faster carrier escape from the wells, thereby improving the diffraction efficiency and resolution of the device (Partovi et a/., 1993). Figure 31(a) shows the absorption spectrum of the material, averaged over the entire MQW thickness. Sharp excitonic features with the heavyhole transition at 847 nm and light-hole transition at 840 nm are observed. Figure 31(b) shows the change in the absorption coefficient of the device with application of a 50-kHz square-wave voltage from 5 to 30V (in steps of 5V) measured with a low-intensity white-light source and an optical multichannel analyzer. Application of ac voltage is necessary to avoid screening of the field from the MQW due to accumulation of photoexcited cariers at the MQW-dielectric interface. Changes in the absorption coefficient of as much as Au e 7500cm-' for 30 V are obtained near the band edge due to the QCSE. The corresponding index change calculated through a Kramers-Kronig analysis is shown in Fig. 31(c). Index changes of up to An z 0.06 are seen. The calculated diffraction efficiency [using the values of Au' = Au/2 and An' = A42 from Fig. 31(b) and Fig. 31(c) in Eq. (72)] is shown in Fig. 31(d). Diffraction efficiencies of several percent are seen over a 10-nm region. Another commonly used definition of diffraction efficiency is the ratio of the diffracted beam to the incident beam qin = r,~,,,e-"~ and takes the insertion loss e - a L into account. The maximum diffraction efficiency qin= 1.5% and occurs at 849 to 850 nm for 30 V applied. For most applications, the signal-to-noise ratio is determined by the ratio of the diffracted signal to the background noise represented by the dc undiffracted beam. Therefore, the diffraction efficiency q,,,, as represented in Fig. 31(d) is more appropriate for optimizing the wavelength of operation. The experimentally measured diffraction efficiency at I = 852.5 nm as a function of applied voltage and grating period is shown in Fig.32. A maximum diffraction efficiency of 3% for an applied voltage of 2 0 V at 30.9-pm grating period is observed. The diffraction efficiency drops very rapidly for smaller grating periods and has a value 3 orders of magnitude smaller for a grating period of 3.3 pm.

5

PHOTOREFRACTIV~TY IN SEMICONDUCTORS

377

4ooo ZDDO

0

.moo .400) .(008

ddoo 0.02 0.01

0

441 -0.02

6.03

om 6.W 4.06

FIG. 31. (a) Measured absorption coefficient without voltage. (b) Change of absorption coefficient Am for a 50-kHz square-wave applied voltage of 5 to 30V in 5-V increments in SI = MQW device. (c) Values for refractive index change An calculated through the KramersKronig relation using the Aa values of (b). (d) The calculated q,,, using Eq. (72). (From Partovi et al., 1993.)

0

g

10

?0 .-

I

-B

0.1

APPLIED ELECTRIC FIELD (kVlan) 150

c

E U

U

Iy

0.01

0

64

P U

0.001

Y o.ooo1 P

h4 1 0

10 20 30 APPLIED VOLTAGE (V)

40

FIG. 32. Output diffraction efficiency as a function of applied voltage for several grating periods at 2 = 852.5 nm. The frequency of the applied ac square-wave voltage is optimized at every data point. The intensity in each beam was 96mW/cmz. The diffraction efficiency saturates to a maximum value of -3% for 20 V. (From Partovi et al., 1993.)

The degradation of the diffraction efficiency at small grating periods is not desirable for most applications and was later improved by incorporation of a high-resistivity LTG AlGaAs layer at the dielectric interface (Bowman et a/., 1994). The MQW used in these improved devices consists of 75 periods of lWA GaAs wells and 35-A AI,Ga, -,As barriers. Devices with x = 0.1 and x = 0.3 were grown. The MQW region was made semi-insulating through proton implantation after sample growth. In addition, both devices contained 300A of AI,,,Ga,,,As material grown at 330°C and annealed at 580°C. The LTG high-resistivity (10l2 0 - c m ) cladding layers reduce diffusion of optically generated carriers at the semiconductor-dielectric interface and result in higher spatial resolution. The resistivity of the M Q W region in these devices was increased by proton implantation. Figure 33 shows the dependence of the diffraction efficiency on grating period for the two samples studied. While the 30% A1 sample showed slightly better diffraction efficiency at large grating periods, both samples maintained superior diffraction at smaller grating periods. In particular, diffraction from the device with 10% A1 drops to one-third its optimal value at A = 2.6 pm. The lower value of maximum diffraction efficiency in these samples compared with the earlier results by samples without the LT layers is caused by the smaller MQW active layer thickness (1 pm compared with 2 pm). The rise time for the grating was as fast as 2 p s for 200 mW/cm2 absorbed intensity (0.4 pJ/cm2 h e n c e necessary for a saturated grating).

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS 0.81

-

...

I

7

7

-

.

I

.

*

-

I

..

.I .

. .

379

.

a7 -

z6 a6 I as-

e

3

a4

c

ii

D

0.3

-

ai n

v

n

-1.

a2-

1 . 4 . .. 0

I

10

A!

-lox

- - ~ - 3 0 % p1

I

. .

.

I

20

. .

.

I

.

30 Grating perbd (Irrn)

. .

I

40

.

.

.

50

FIG. 33. Grating period dependence of the diffraction efficiency signal for 10% and 30%A1 barrier devices. (From Bowman et ol., 1994.)

The large improvement of performance with incorporation of LT layers led to design of devices with the LT layers at the interface and intrinsic MQWs as the highly nonlinear medium. Figure 34 compares the diffraction efficiency for devices with 30% A1 MQW as described earlier and LT barriers with ion implantation to make the MQW semi-insulating (SI/LT), without the ion implantation (INT/LT), and a device made with a material containing similar MQWs but no LT barriers and no ion implantation (INT). Similar performance was obtained for the two devices with LT

3

FIG. 34. Diffraction efficiency dependence on grating period for semi-insulating/low-temperature-grown (SILT), without ion-implanted (INTLT), and without low-temperaturegrowth (INT) devices. (From Rabinovich et al., 1995.)

JAMESE. MILLERD,MEHRDADZIARIAND AFSHINPARTOVI

380

barriers. Thus, the ion-implantation step is not necessary for achieving high diffraction efficiency and resolution. The device without the LT barriers, however, had a diffraction efficiency that was reduced by an order of magnitude from the other two devices. The reason for the importance of the interface LT material is that the carriers optically generated in the MQW very rapidly (several picoseconds) drift to the interface region to accumulate and screen the electric field from the MQW. Reduction of carrier diffusion at these locations is very important for maintaining high fringe visibility and diffraction. An in-depth study of picosecond time-resolved diffraction from samples with dielectric buffers also points to the trapping of charges at the MQW-buffer as the main factor in determining the spatial resolution of PR-MQW devices (Canoglu et a/., 1996). To estimate the fluence F required to write a grating, we note that to screen a field E across a material with thickness L, the density of carriers required can be written as

where

d N Qa21 -=dt

hv

is the carrier generation rate at the intensity maxima (= 21) for a sinusoidal intensity pattern with two equal beams of intensity 112, and z is the response time. Q is the quantum efficiency, defined as the ratio of the photogenerated carriers that reach the semiconductor-dielectric interface, and hv is the photon energy. Using the appropriate values for the samples used and E = 100 kV/cm, we obtain QIra = 0.022 pJ/cm2. The absorbed fluence necessary to write a saturated grating for the INT/LT sample was 0.08 pJ/ cm2, which is much smaller than the value for the SI/LT sample (-0.35 pJ/ cm2). This means that the INTILT sample requires 4 photons to produce one e-h pair worth of screening, while the SI/LT sample requires 16 photons. With no traps in the M Q W region, the Q of the device is much higher because the carriers are much less likely to get trapped during their transport to the interface.

2.

ELIMINATION OF THE DEWSITED LAYERSAND SUBSTRATE REMOVALIN PR-MQWs

Although the PR-MQW devices described so far show high diffraction efficiencies and sensitivities, they suffer from the problem that their fabrica-

5

PHOTOREFRACTIVITY IN SEMICONDUCTORS

381

tion involves many etching and deposition steps, increasing the cost and complexity of the device. Devices that would eliminate the deposited dielectric/transport electrodes and the substrate-removal step have been proposed and demonstrated in the last several years (Haas et al., 1995; Partovi, 1995; Pelekanos et al., 1995; Lahiri et al., 1995, 1996). Lahiri et al. (1996) have demonstrated devices consisting of 2pm LTG shallow MQW material placed between 5000-A AI,,Ga,.,As cladding layers and p- and n-doped contact layers. Shallow MQWs provide high-mobility vertical transport, fast escape of photo-carriers, and enhanced electro-optic sensitivity, which are all desirable qualities. Devices with the buffer layers grown at LT (320°C) and high temperature (600 and 450°C for the lower and top layers, respectively) were fabricated. In the first type (with LTG buffers) the buffer acts as a trapping layer, while in the second kind the buffers simply act as standoff layers. Figure 35 shows the transient output diffraction efficiency as a function of the electric field for both devices for a single-sided square pulse at 100 Hz repetition rate. A record qout= 35% was obtained for 20% duty-cycle pulse for the sample with the standoff. The researchers speculate that the increase of qout at low duty cycle is caused by the nonlinear dependence of the diffraction efficiency on internal fields. The sample with the intrinsic standoff has lower voltage dropped across the buffer, allowing more voltage to drop across the MQW layer compared with the LTG buffer device. An important design flexibility offered by the growth capabilities of thin epitaxial films is the possibility of design of high-quality Fabry-Perot etalons to increase the signal contrast ratio. MQW modulators using asymmetric Fabry-Ptrots are designed for the reflected light from the front interface and a high-reflectivity rear reflector to be out of phase. By designing the amplitude of the two components to be equal, the reflected light intensity is minimized and is highly sensitive to changes in the dielectric properties of the MQW. The largest contrast ratio MQW modulators are based on such asymmetric Fabry-PCrot devices (Yan et al., 1990; Law et al., 1990). This is discussed in greater depth in Chap. 2. Application of Fabry-Ptrot designs to MQW-PR devices has been theoretically analyzed (Nolte and Kwolek, 1995; Nolte, 1994). Photorefractive MQW p-i-n devices grown on a quarter-wave stack n-type mirror with LTG buffer layers and 1.5 pm of ultra-narrow barrier intrinsic MQW have been fabricated (Tayebati et al., 1997). After growth, the processing of these devices only consists of etching the p layer in an area around the periphery of the device down to the LTG layer, metallizing the contact pads and connecting electrodes. Figure 36(a) shows the qoutfor this device, measured in reflection as a function of wavelength and incidence angle at 30 V reverse applied voltage. These devices were not intentionally designed to behave as

382

JAMES E. MILLERD,MEHRDADZIARIA N D AFSHIN PmTovi I

.

.

.

l

'

.

.

I

.

0 A

2

.

.

,

.

.

.

I

stand-off (50%:. Buffer(5096)

4 6 8 Electric Field (v/pm)

10

FIG. 35. Output diffraction efficiency as a function of the electric field for the buffered and standoff device with an electric field repetition rate of 100 Hz with 50% and 20%' duty cycle at A = 18.5pm.(a) The transient time response (TTR) is the peak diffraction efficiency after switching of the electric field, while (b) is the time-integrated response (TIR), which is the average of the diffraction efficiency over one period of the electric field. (From Lahiri et a/., 1996.)

asymmetric Fabry-Perots. However, the interference in this thin device creates a highly angle-dependent g. The maximum go", = 0.5% at 60-degree incident angle at 845.5nm coincides with a sharp drop in zero-field reflectivity, as shown in Fig.3qb). A maximum q,., of 2% a t 30-degree incident writing angle for -40 V was measured for another sample from the same wafer. One drawback of the QCSE-MQW geometry is that the intensity pattern and the grating are in phase. Therefore, energy transfer between two writing beams does not occur. Using moving gratings by shifting the frequency of one of the beams using an A 0 modulator, gains approaching 1OOOcm-' have been demonstrated in PR-MQW p-i-n devices (Lahiri et al., 1996).

PHOTOREFRACTIVITY IN SEMICONDUCTORS

5

I

1 8

383

0.1 0.01

0.001

I,-I

I-. . .

. I . .

845

. . ..-. I

850

.

I . .

855

. . I .

.J

860

W8velength [nml

W8velengtb [nm]

FIG. 36. (a) Wavelength dependence of the diffraction efficiency for -32 V applied voltage at different angles of incidence and (b) corresponding reflectance spectra. (From Tayebati et al., 1997.)

3. MQW-PR USINGTHE FRANZ-KELDYSH EFFECT For photorefractive applications, a geometry where the field is applied parallel to the MQW layers (the geometry that utilizes the Franz-Keldysh effect to create the grating) has the advantage that since the charge transport and the space-charge buildup in the material are through drift and diffusion similar to bulk materials, a phase shift between the intensity pattern and the resulting grating exists. This phase shift allows energy transfer between writing beams (beam coupling) to occur. The other advantage of this geometry is the simplicity in fabrication of the device. It is only necessary to apply contacts on one of the surfaces of a thinned SI-MQW material and apply an ac or dc voltage to the contacts. In comparison, the devices in the QCSE geometry require more complicated structures requiring buffer layers and conducting transparent electrodes. However, as described in the last

JAMES E. MILLERD.MEHRDADZlARI AND AFSHINPARTOVI

384

section, recent devices where these layers are epitaxially grown eliminate the need for most of the postgrowth processing. The first demonstrations of the PR-MQW effect in MQWs were carried out in the F-K geometry (Glass et al., 1990, Nolte et al., 1990b; Wang et al., 1991, 1992). Nondegenerate four-wave mixing diffraction efficiencies of 3x and beam-coupling gain coefficients of 900cm-' have been obtained for an ion-implanted SI-MWQ sample (Wang et al., 1992). To increase the diffraction eaciency, nearly symmetric Fabry-Perot samples with 1 pm MQW thickness were designed (Kwolek et al., 1994). Maximum output diffraction efficiencies of 1.3% in reflection and 0.7% in transmission at A = 26 pm, E = 9 kV/cm were obtained. Operation in reflection allows the beam to travel through the media multiple times, thereby increasing the effective sample length and diffraction efficiency. Design of asymmetric Fabry-Ptrot devices with high (> 95%) backmirror reflectivity results in devices with very high input and output q (Kwolek et a/., 1995). Figure 37 shows results for standard-temperaturegrowth (STG) and low-temperature-growth (LTG) (T = 320°C) samples. The two LTG devices differ from each other only in their buffer thickness and therefore in their Fabry-Pirot fringe position. The STG sample was made semi-insulating by ion implantation, while only a 30-s anneal at 600°C is needed for the LTG sample to become semi-insulating through arsenic precipitate formation. A record q,,, = 200% for a perfectly balanced cavity was obtained. The corresponding qin was 0.12%.

-

845

847.5

850 852.5 Wavelength (nm)

855

FIG. 37. Output diffraction efficiency for three samples of asymmetric Fabry-PCrot devices. STG, standard-temperature growth; LTG, low-temperature growth. Corresponding peak input diffraction efficiencies are also shown. (From Kwolek et al., 1994.)

5

F’HOTOREFRACTIVIIT IN SEMICONDUCTORS

385

VIII. Selected Applications Many applications for photorefractive semiconductors (both bulk and quantum well) are actively being investigated by industry and academia. Several of the applications are presented in the following subsections.

1. COHERENT SIGNAL DETECTION (ADAPTIVE INTERFEROMETER) The two-beam coupling process in photorefractive semiconductors can be thought of as an adaptive interferometer because light from the pump beam couples to the signal beam exactly in phase (Davidson et al., 1988). If the signal wave has a spatially varying phase and amplitude (as may occur from reflection from a diffuse surface), the hologram in the crystal will exactly compensate for the variations, and the pump beam will diffract everywhere in phase with the signal beam. The ability to dynamically combine two beams that have arbitrary wavefronts has important applications in coherent signal detection. A coherent signal processor consists of a reference and a signal beam that are combined to produce a beat signal on a detector, as shown in Fig. 38. The signal beam usually has time-varying phase information encoded on it (e.g., phase modulation for data communication, frequency shift from a vibrating target). Detecting the beat signal on the detector allows the phase modulation to be decoded. To achieve the highest possible signal, the wavefronts from the reference and signal beams must be exactly matched and overlapped. Spatial filtering can be used to ensure good overlap, but in the case of complex wavefronts, much of the energy is discarded.

Phase ‘hase

nal reference W

B

cl Detector

S

FIG. 38. Conventional coherent signal processor (Mach-Zender interferometer).

386

JAMES E. MILLERD,MEHRDADZIARIAND AFSHINPARTOVI

The adaptive interferometer of Fig. 39 employs a photorefractive crystal to effectively compensate for any wavefront mismatch between the reference and aberrated signal beams. Low-frequency fluctuations on either beam (f < l/.c,) will be dynamically compensated for by the grating in the photorefractive crystal; however, the crystal does not respond fast enough to compensate for high-frequency phase fluctuations in the signal beam (f >> I/rp). Therefore, the photorefractive adaptive interferometer performs two functions: It matches wavefronts between the reference and signal beams, and it acts as a high-pass filter (Blouin and Monchalin, 1994). Ultrasound detection is one application where the adaptive interferometer is particularly well suited. Ultrasound can be used as a manufacturing inspection technique to detect cracks and corrosion in critical components. Conventional ultrasound measurement employs mechanical contact sensors; however, a laser beam can be used to remotely detect high-frequency vibration of a target using interferometry. The adaptive interferometer is ideal for this situation because the probe beam emerges as a highly aberrated and speckled beam from most work surfaces, as shown in Fig. 39. Provided the magnitude of the phase modulation on the signal beam is very small, A+ -K a (equivalent to a small vibrational amplitude), a grating will be written in the photorefractive crystal that precisely overlaps the diffracted reference beam with the signal beam. By selecting an appropriate photorefractive time constant, the adaptive interferometer can compensate for low-frequency phase fluctuations, such as bulk motion between the target and the sensor (typically < kHz), yet the high-frequency ultrasound component (typically > MHz) will be passed on to the detector. Careful analysis of the signal reaching the detector shows that for diffusion-type recordings (no applied field, Ed, = 0), the signal reaching the detector is proportional to the square of the phase modulation, I, cc Acp2(t)

-

LASER

signal BS reference

"\

TI I

FIG. 39. Adaptive interferometeremploying a photorefractive semiconductor (PRSC).

5

PHOTOREFRACTIVITY IN SEMICONDUCTORS

387

(Delaye et al., 1995). This can be viewed as a consequence of the fact that the twer-Beams (reference and signal) add in phase. If the grating can be spatially shifted to 0 or 180 degrees, the two signals will add in quadrature, and the signal leaving the crystal will be linear with the phase modulation, I , K A&). Using externally applied electric fields and/or moving gratings, the phase of the grating in photorefractive semiconductors can be adjusted to meet this condtion (see Eq. 9). This application of photorefractive semiconductors is currently being explored by several major research groups (Pepper, 1997).

2. OPTICAL IMAGEPROCESSING Holography can be used to perform a wide variety of image-processing functions such as spatial filtering, correlation, and image transfer (Goodman, 1968). The fast holographic response of photorefractive semiconductors at the wavelengths of near-infrared semiconductor lasers makes them ideal for real-time holographic image processing. Image transfer is a straightforward process using four-wave mixing and is shown in Fig. 40(a). A transparency is placed in the signal beam while the reference and reconstruction beams are counterpropagated and collimated. The diffracted reconstruction beam is a phase-conjugate replica of the signal beam, which will reimage at original transparency. A beamsplitter is used to divert the phase-conjugate image. Figure 40(b) shows a phase-conjugate image of a test resolution pattern produced using ZnTe:V:Mn in a four-wave configuration and recorded on a CCD camera. The reconstruction beam can come from a different laser with a different wavelength, and hence image transfer

Keferenc

Reconstruction

x:onstructed image

FIG.40. Four-wave mixing configuration used for large transfer and a phase-conjugate image of a test pattern produced using ZnTe at 765 nm.

388

J

m E. MILLERD,MEHRDADZURI AND AFSHINPARTOVI

between different optical beams is accomplished. Also, the phase-conjugate nature of the reconstructed image exactly compensates for any phase nonuniformities in the photorefractive crystal or between the original image and the crystal. By introducing a lens between the transparency and the crystal and adjusting the relative intensities of the beams, the image can be spatially filtered. For example, edge enhancement can be accomplished by adjusting the intensity of the reference beam to be equal to high-order spatial frequency components. The stronger-intensity low-frequency components will write a grating of smaller modulation index and diffract less light. Thus the low spatial frequencies will be suppressed (Gheen and Cheng, 1987).

3. OPTICAL CORRELATORS The four-wave mixing arrangement also can be used to perform optical correlations. By passing the reference beam through an image (rather than using a uniform planewave) and by focusing both reference and signal beams into the sample, the diffracted light will be proportional to the joint transform correlation of two images. Likewise, placing an image in the reconstruction beam (and nothing in the reference beam) will create a VanderLugt optical correlator (Liu and Cheng, 1992). The optical layout for the VanderLugt and joint transform correlators are shown in Fig. 41. The VanderLugt and joint transform correlators are both useful for image recognition and authentication. In addition to amplitude information, phase objects also may be correlated. The combination of the correlator and the spatial filter can be used to perform nonlinear correlation and enhance the signal-to-noiseof the correlation signal (Javadi and Homer, 1994). Uses for optical correlators include industrial recognition (e.g., distinguishing various

FIG. 41. (a) Joint transform and (b) VanderLugt photorefractive optical correlators.

5

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389

washers and nuts in a factory), hand-written character and signature recognition, fingerprint recognition, security authentication of credit cards, currency, and badges, medical applications including the identification of cells (Pepper, 1997), and data bandwidth reduction (Stark, 1982). Correlators have been demonstrated using bulk GaAs that have correlation times of milliseconds (Liu and Cheng, 1992). In order to achieve sizable (> 1%)diffraction efficiency in bulk photorefractive materials, samples with a thickness of 1 to 5mm are required. Such thick holograms operating in the Bragg regime severely restrict the space-bandwidth product (number of resolvable spots) in the input images of these correlators (Gheen and Cheng, 1988; Athale and Raj, 1992). The thin nature of MQW-PR devices offers an advantage over bulk materials, increasing the space-bandwidth product (SBWP) for optical correlator systems. MQW-PR devices are capable of sensitive (<0.1 p.J/cm2) and fast (< 1ps) operation at diode laser wavelengths (- 850 nm). Diffraction efficiencies of several percent in the first Raman-Nath order have been obtained in devices with 1 to 2 pm thickness. Optical image (Pepper et al., 1978; White and Yariv, 1980; Rajbenbach et al., 1992) and rf-signal correlators (Hong and Chang, 1991) have been implemented with MQW-PR devices. Figure 42 shows a joint Fourier transform optical image correlator that used two diode lasers and a GaAsIAlGaAs SI-MQW device as the real-time holographic element. In this implementation, the images were input to the system using a liquid-crystal spatial light modulator (LCSLM), and the

Liquid Crystal SLM Polarizing 1 Beam-Splitter

8 Band

PR-MQW Device

B V D C Beam Block

"

FIG. 42. Schematic diagram of the experimental setup for the joint Fourier transform correlator. The images to be correlated are input to the system using a liquid-crystal SLM (shown on the left). The images are Fourier transformed by a lens, and the resulting pattern on the PR-MQW device causes a second laser diode beam at 8Mnm to diffract off of the pattern and form the correlation pattern on the back focal plane of the lens. (From Partovi et nl., 1993.)

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AFSHIN PARTOVI

output was captured by a CCD camera. The time for a single correlation was 1 p s (Partovi et al., 1993). Figure 43 shows a surface plot of a portion of the output of the CCD camera in response to the hand-written input image shown in Fig. 43(b). A strong peak corresponding to the autocorrelation of the bottom “2” with the similar upper right “2” and a weaker cross-correlation peak for the bottom “2” with the different upper left “2” is seen. As expected, no strong correlation peaks are observed for the digit “4.” This example shows the high signal-to-noise ratio achievable with SI-MQW correlators. The lack of scatter noise is a major advantage of atomically flat MQW devices. In the correlator system of Fig.42, the system throughput is limited by the input spatial light modulator and the output CCD camera to 30 frames/s. To fully take advantage of the microsecond response time of the SI-MQW device, faster SLMs and a method for data reduction at the output detection plane are required. The gray-scale response of the diffracted beam from PR-MQWs has been shown to depend on the incident intensity, the applied voltage, and frequency (Rabinovich et al., 1996). In correlator applications, it is therefore possible to perform high-pass or low-pass filtering of the input images by tuning the parameters to preferentially diffract the desired components in the Fourier plane at the PR-MQW. Such a nonlinear correlator allows more versatility in system performance and improves the correlation. The same principles used in the correlation of images can be used to measure electric field autocorrelation and cross-correlation of femtosecond

FG. 43. (a) Surface plot of a portion of the output of the CCD camera in response to an input image shown in (b) consisting of the hand-written number “242” to be correlated with “2.” (From Partovi el uf., 1993.)

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FIG. 44. Experimental setup for space-to-time conversion and pulse shaping of femtosecond pulses. (From Ding et al., 1997.)

pulses in real time (Nuss et al., 1994). In this case, a diffraction grating is used to disperse the short temporal pulses to their spectral components in space. Using these dispersed beams as input to a correlator similar to the one described earlier and reading out with a CW diode laser, the correlation signal is obtained as a spatial image on a CCD. Figure 44 shows an experimental setup used for real-time modification of femtosecond pulses (Ding et al., 1997). In this case, an incident femtosecond pulse is dispersed by a grating, and the spectral components are focused by a cylindrical lens onto a PR-MQW device. A similar lens collects the beam on the opposite side of the PR-MQW, and a grating recombines the spectral components to produce a single temporal pulse. Using the PR-MQW as an optically addressed SLM in the Fourier plane, the spectral components of the pulse can be modified to produce pulse-shaping effects such as edge enhancement in the time domain. This could be useful for separating closely spaced pulses of light in optical communications. 4.

REAL-TIME HOLOGRAPHIC INTERFEROMETRY

Holographic nondestructive testing (HNDT) is a proven diagnostic tool for detecting component flaws and performing modal analysis of static and rotating machinery. Photorefractive semiconductors can replace conven-

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E. MILLFXD.MEHRDADZURl AND AFSHINPARTOVI

tional holographic films and greatly increase the speed and utility of HNDT by providing real-time holographic interferogram. Time-averaged holography can identify resonant modes and quantitatively measure vibrational displacements of parts and structures (Petrov et at., 1991). Double-pulse holography can be used to measure time-dependent flow fields for aerodynamic and combustion applications (Magnusson et at., 1994). And twocolor holography can be used to measure the three-dimensional surface profile of optically rough surfaces. (Kuchel and Tiziani, 1981). Many of these techniques can greatly benefit from photorefractive semiconductors that require no chemical processing, allow real-time readout, and are compatible with laser diodes. One emerging holographic application that can particularly benefit from the fast response time of photorefractive semiconductors is resonant holographic interferometry (RHI) (Dreiden et at., 1975; Rubin and Swain, 1991). RHI provides a method for obtaining species-specific interferograms by recording two simultaneous holograms at two different wavelengths, one tuned near a chemical absorption feature and the other tuned off this feature (typically < 0.1 nm separation between lasers). Since phase contributions to the interferogram from background species, thermal and pressure gradients, and optical aberrations are subtrated out in the holographic reconstruction process, the resulting interference fringes correspond uniquely to the density of the species under interrogation. The interferogram permits two-dimensional chemical detection that is useful for combustion and plasma diagnostics, medical imaging, and flow visualization. The use of photorefractive semiconductors extends the current RHI technology into the near-infrared (NIR) spectral region, where conventional holographic media are unavailable, and enable real-time measurement capability. The NIR region is easily reached with inexpensive, commercially available laser diodes. In addition, semiconductor materials can have photorefractive response times of picoseconds, so they offer the potential for ultra-high-speed data acquisition (Valley et a/., 1989). The optical layout for a real-time RHI system is shown in Fig.45. The laser beams tuned to the on- and off-resonance wavelengths are s-polarized, pass collinearly through the test object (combustion flowfield, plasma reaction chamber, etc.), and are focused to match the crystal aperture. The two reference beams, at the same wavelength, are also collinear and s-polarized. Reconstructed phase-conjugate replicas of both beams produce interference fringes at the detector plane that correspond to the density of the species under interrogation. The polarization rotation properties of four-wave mixing in semiconductor photorefractive crystals may be used to suppress noise from scattering (Yeh, 1987). The photorefractive crystal is oriented so that the diffracted

5 PHOTOREFRACTIVITY IN SEMICONDUCTORS

Recmstmtedphase map

393

CCD camera

of tesonaMx species

*

Image plane

Testobject ’

FIG.45. Phase-conjugate configuration with polarization switching or real-time resonant holographic interferometry system. (From Millerd er al., 1996.)

signal of the counterpropagating reconstruction beam has its polarization rotated by 90 degrees. The polarizing beam splitter (PBS) increases the signal-to-noise ratio by selecting only the diffracted light and rejecting any s-polarized scattered light. Additionally, the reconstruction beam can be a different wavelength than the other beams, and chromatic filtering can be used to attenuate scattered light. ZnTe:VMn has been used to record real-time RHI interferograms using the configuration shown in Fig.45. Pulsed YAG pumped-dye lasers were tuned to 0.15 nm on either side of the D, absorption feature. Laser energies at the crystal were lmJ/pulse. Single droplets on the order of l m m in diameter of KCI mixed in methanol were suspended from a fine wire and ignited with a butane flame. After ignition, the butane flame was removed, and the droplet was allowed to burn on its own. The evolution of the combustion process was monitored at 1 0 H z with the real-time RHI instrument (Millerd et d.,1996). Figure 46 shows a sequence of successive RHI interferograms of a KCIseeded, burning methanol droplet. The images are separated by 100-ms intervals (10Hz). Figure 46(a) shows the droplet just after ignition. The absence of fringes due to thermal gradients highlights the utility of holographic optical background subtraction. In subsequent frames, fringes are

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JAMB E. MILLERD.MEHRDAD ZIARIAND AFSHIN PARTOVI

FIG. 46. RHI interferograms of burning methanol droplets seeded with KCI. Each image was recorded on a CCD and is separated temporally by 100ms. (From Millerd er al., 1996.)

clearly visible. These images may reveal many of the finer details of the flame front development and demonstrate that real-time RHI, using photorefractive semiconductors, is a promising diagnostic for studying combusting droplets or other multiphase, highly luminous, highly scattering events. The list of applications presented here is not comprehensive, and many other uses are undoubtedly still yet to be discovered. The wide possibilities of substitutional dopants, alloying, and atomic-layer growth techniques leave a vast potential for future research. Both the fundamental understanding of solid-state physics and applications development are likely to motivate photorefractive semiconductor research in the future.

LIST OF ABBREVIATIONS AND ACRONYMS HTO (‘W

DFWM DPCM EO

Bi 12Ti020 continuous wave degenerate four-wave mixing double-pumped phase-conjugate mirror electro-optic

5 EOPR ER ERPR F-K HNDT LTG MBE MQW NIR PR QCSE RHI SI SPCM STG

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395

electreoptic photorefractivity electrorefractive electrorefractivephotorefractivity Frank-Keldysh holographic nondestructive testing low-temperature growth molecular beam epitaxy multiple quantum wells near-infrared photorefractivity quantum confined Stark effect resonant holographic interferometry semi-insulating self-pumped phase-conjugate mirror standard temperature growth

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SEMICONDUCTORSAND SEMIMETAL$ VOL. 58

Index Numbers followed by the letter f indicate figures; numbers followed by the letter t indicate tables.

A

figures of merit regarding, 34-36,37t intersubband absorption saturation, 31 in optical modulators and active media, 32-33 quantum dots, 32 in quantum wells, 29, 3Ot, 31 response time of, 35-36 Band-gap renormalization, 203 Band structure, anisotropy of, 283-285 Band gap, renormalization of, 6 Bare Coulomb interaction, 214 Beam coupling equations of, 328-330 intensity dependence of, 332-333 spatial frequency dependence of, 330-332 speed of, 333-335 Beam distortion, 295-296 Bloch equations, semiconductor, 214 Bloch oscillations, 224 Bloch's theorem, 181 Blocking factor, 3-5 Bound-electronicnonlinearities, 285-287 two-band model of, 271-285 ultrafast optical switching using, 308-309 B r a g mirrors, 139- 140 Fabry-Perot resonators using, 144- 145, 157- 158 nonlinear, 145-146, 159 reflectivity of, 140-143 Brillouin zone, 178 Bulk semiconductors anisotropic band structure of, 283-285 band-filling nonlinearities in, 29, 30t, 31

Absorption density-dependent, 19-25 linear, 18-19 measurement of, 19-25,27-29 nonlinear, 14- 16 Absorption coefficient, nonlinear, 34 Absorption experiments, 126-128 Absorption gratings, 329 Absorption lines, 2-3 broadening of, 6-7 AC fields, in wave mixing, 348-350, 363t AC Stark effect, 271 Adaptive interferometer, 385 All-optical switching, 45 demonstrationsof, 47,49 devices for, 48t mechanisms of, 45-47 using bound-electronic nonlinearities, 308-309 AND gates, optical, 46 Anisotropic strain, effect on QWs, 110-1 11 Asymmetric quantum wells, 103-104

B Bhbyai-Koch model, I2 Band-edge electrorefraction photorefractivity (ERPR),355 Band-edge enhancement, 352-354 Band filling, 3, 8 Band-filling nonlinearities in bulk semiconductors, 29.3Ot, 31

403

404

INDEX

Bulk semiconductors (continued) boundelectronic nonlinearities in, 285287,308-309 experimental techniques for, 293-31 1 freecarrier nonlinearities in, 287-293 materials for, 365-371 nonlinear absorption in, 259-260,271277 nonlinear polarization in, 261 -271 nonlinear refraction in, 259-260,277283 performance of, 368t photorefractive sensitivity of, 364

modulator, 64-65 Dielectric Constant, Drude model for, 38 Dielectric relaxation rate, 334 Diffusion, lateral enhanced, 155- 156 Diffusion length, 334 Diffusion time, 335 Direct-bandgap semiconductors, absorption edge Of, 2-3 Drude contribution, 290 Drude model, 38 Dynamic controlled truncation scheme (DCTS), 229-230

E C

Carrier transport enhancing nonlinearities, 63-85 and local nonlinearities, 57-60 Cavity finesse, 15 CdMnTe, 369 characteristicsof, 368t, 370 CdS, characteristicsof, 369,368t CdSe, 371 CdTe, 371 characteristics of, 367 CdTe:V, characteristicsof, 368t. 370 CdZnTe, 369,370 Coherent signal detection, photorefractive semiconductors in, 385-387 Compressive strain, effect on QWs, 109 Contrast ratio, 123 Coulomb interaction, 4-5 screening of, 6 Coupled quantum wells, 79-80, 103-107 strained, 113 Coupled-mode theory, 143 Coupled-waveequations, 328-330 Cross-polarization coupling, 344-346 D

DC fields, in wave mixing, 346-347 Degenerate four-wave mixing (DFWM), 297-298, 338-340 Densitydependent absorption, 19-21 measurement of, 19-20 modeling of, 23-25 Depletion region electroabsorption

Electro-optic effect, 326-327 Electron-hole interactions, 336-337 Electron-hole pairs, 2 Electrorefraction, 100- I02 Elliott formula, 182 Energy bands filling of, 3, 8 saturation levels for, 11 Enhanced diffusion, 155- 156 Etalons, 138- 139 Excitation-induced dephasing (EID), 232 Excitation-probe measurements, 296-297 Excite-probeZ-scan, 302-303, 305 Exciton behavior of, 184-185 defined, 4-6, 180 and saturation density, 10-1 1 Exciton oscillator strength, 207 Excitonic optical start effect, 198-206

F Fabry-Perot cavity, nonlinear, 14, 15f, 45-47 Fabry-Perot geometries, 128- 129 reflectivity of, 129-130 Fabry-Perot resonators, 138-139 using B r a g mirrors, 144- 145, 157- 158 Femtosecond continuum probe, 305-306 Fermi edge singularity (FES), 187 Fermi golden rule, 193 Field shielding, 349 Figures of merit determining, 134- 136

405

INDEX for band-filling nonlinearities, 34-36,37t for optical nonlinearities, 61-62 Four-particle correlation effects, 226-239 Four-wave mixing (FWM), 146,210-211, 223f, 337-338 degenerate, 297-298,338-340 determining nonlinear response, 297-299 diffraction efficiency of, 340-341 enhanced methods of, 346-364 polarization switching, 344-346 self-pumped phase conjugation, 342-344 Fourier transform optical correlator, 389390.389f Franz-Keldysh effect, 74, 354,3569 3571 experiments on, 89-90 MQWs using, 383-384 self-modulation of, 81-83 theory of, 88-89 Free-camer absorption, 38-39 causing optical nonlinearity, 7, 8 Free-carrier gratings, 327 Free-camer nonlinearities, 7,8,39-41,42t, 287-293 Free-carrier refraction, causing optical nonlinearity, 7, 8 Frolich LO-phonon-carrier interaction, 192

HgCdTe, 369 HgTe, 369 Holographic interferometry, 391-394 Holographic nondestructive testing (HNDT), 391-392 Holography. 387-388

I 11-VI alloys, as semiconductors, 369 Image processing, semiconductors in, 387388 Impedance match, 130-134 InP, characteristicsof, 366 InPFe. characteristics of, 368t Insertion loss, 123 Intensity-dependent absorption, 29t measurement of, 25.27-29 modeling of, 25-26 saturation parameters of, 26-27 Interferometry, 299 absorption-only, 126- 127 adaptive, 385 based on phase shift, 127-128 holographic, 391-394 Intersubband absorption saturation, 3 1 J

GaAs, characteristicsof, 365,368t GaAs:Cr, characteristicsof, 368t Gap, characteristics of, 367,368t Glasses, doped, 32 Gratings absorption, 329 moving, 350-352, 363t space-charge, 321-327 spacing of, 360

Joint density of states, 179 Joint transform optical correlator, 388, 3881

K Kane energy, 273 Kramers-Kronig relations, 12- 13.90-91, 264-265 linear, 265-267 nonlinear, 267-271

H Hartree-Fock treatment, 201 Heisenberg equations of motion, 191 Hetero-nipi structures defined, 63 speed of, 85 type I, 70-74, 80 type 11, 70, 74-79, 80-81 HF/RPA meanfield theory, 230

L Lateral enhanced diffusion, 155-156 Linear electro-optic effect, 326-327 Liouville equation of motion, 201 Local nonlinearities defined, 57 enhanced by carrier transport, 57-60 Lorentz contribution, 290

406

INDEX

M Markovian subsystems, 191 Microwaves, optical switching of, 41, 43-44 Modulating doping superlattices, 65. See also Nipi structures Moving gratings, 350-352, 363t M ultidefect interactions, 336-337 Multiple quantum wells (MQWs), 371 characteristics of, 380-383 geometries for, 372-373, 372f substrate removal for, 381 using Franz-Keldysh effect, 383-384 using quantum confined Stark effect, 373-380

N Near-band-edgeeffects, 354-361 Near-band-gap excitations, 178- 188 Nipi structures defined, 63 described, 65-66 intensity dependence of absorption of, 149- 153 lateral enhanced diffusion in, 155- 156 modeling of, 149- 155 optical nonlinearity of, 153- 154 optical properties of, 67-70 performance of, 1 56- 160 picosecond excitation of, 154- 155 Speed Of, 84, 147- 149 Non-Markovian behavior, 240-245 Nondegenerate interaction, 259 Nondegenerate nonlinear absorption, 271-277 Nondegenerate nonlinear refraction, 277-283 Nonlinear absorption, 259-260, 271-277 coefficient of, 34 Nonlinear Bragg mirrors, 145- 146 Nonlinear polarization, 261-271 Nonlinear refraction, 259-260, 277-283 coefficient of, 34-35 Nonlinear self-action, 259 Nonlocal nonlinearities, defined, 60 NOR gates. optical, 46

0

Occupation numbers, 189 [ I l l ] planes, 111-113 Optical computer, 45 Optical correlators, 388-391 Optical image processing, semiconductors in, 387-388 Optical intensity, 9- 10 Optical limiting, 309-31 I Optical modulators, 32-33 Optical nonlinearities based on state filling, 64-74 equations governing, 38-39 figures of merit for, 61-62 free carrier, 39-41.42t historical studies of, 258-259 local. See Local nonlinearities nonlocal. See Nonlocal nonlinearities optothermal, 44-45 resonant. See Resonant optical nonlinearity simulations of, 153- 154 thin film epitaxial geometry for, 56-57, 57f Optical start effect (OSE), 198-206 Optoelectronics, use of semiconductors in, 218-226 Optothermal nonlinearities, 7.44-45

P Passive optical limiting, 309-31 1 Pauli principle, 183 Phase space filling (PSF), 3, 184 Phonon oscillations, 245-246 Phonon scattering, 192-198 Photocarrier density, 9, 58 Photonic switching, 45 Photorefractivedevices using Franz-Keldysh effect, 383-384 using quantum confined Stark effect, 373-380 Photorefractive effect, 320- 32 1 and beam coupling, 328-337 at high modulation depths, 361-363 models of, 321-324 steady-state view of, 324-325 Plane-wave interference model, 321-323 Plasma gratings, 327

INDEX Pockels electro-optic photoreactivity (EOPR), 355 Polarization effects of, 283-285 nonlinear, 261-271 total material, 261 Polarization amplitudes, 189 Polarization interference, 219-220, 221f Polarization scattering, 192 Polarization switching, 344-346 Pumpprobe measurements, 296-297

Q

Quadratic optical Stark effect (QSE), 272 Quantum beats, 219-220,221f Quantum confined Stark effect (QCSE), 75, 91-95

in asymmetric quantum wells, 104 dependence on well characteristics, 96-99 and MQWs, 373-380 at 980nm, 114-1 17 at 1.06pm, 117-118 at 1.3pm, 118-120 at 1.55pm, 120-122 in strained quantum wells, 108-1 13 in symmetric quantum wells, 103-104 at visible wavelengths, 113- I 14 Quantum dots, 32 Quantum kinetics non-Markovian behavior in, 240-245 phonon oscillations, 245-246 uncertainties about, 247-248 Quantum wells (QWs), 7 asymmetric, 104 band-filling nonlinearities in, 29, 301, 31 coupled, 79-80, 103-107, 113 density-dependent absorption of, 19-25 depth of, 98-99 design and fabrication of, 16- 18 electrorefraction in, 100- 102 excitonic resonance characteristics of, 93-95

intensity-dependent absorption of, 25-29 linear optical absorption of, 18-19 multiple, 371-384 polarization dependence of, 102 refractive index measurement of, 21-23 stepped, 106 strain on, 108-113 symmetric, 103-104

407

width of, 96-98 Quantum wires,32

R Rabi frequency, 201 Raman contributions, 271,273 Refraction coefficient, 34-35 nonlinear, 34-35 Refractive index, 136- 137 measurement of, 21-23 Resonant effects, causing optical nonlinearity, 7-8 Resonant holographic interferometry, 392394, 3931

Resonant optical nonlinearity, 2 measurement of, 8-16 in quantum wells, 16-29 types Of, 3-8

S Saturation density, 8, 10-1 1 Saturation intensity, 8-9, 65 Schriidinger equation, 181 Self-modulation devices using, 146- 159 ofmultiple junctions, 83-85 of single junction, 81-83 Self-pumped phase conjugate mirror, 342 Self-pumped phase. conjugation, 342-344 Semiconductor amplifiers, band-filling nonlinearities in, 33 Semiconductor Bloch equations, 214 Semiconductordoped glasses, 32 Semiconductor etalons, 138- 139 Semiconductor lasers, band-filling nonlinearities in, 33 Semiconductors absorption spectra of, 86-87 applications of, 385-394 band-filling nonlinearities in, 29, 30t, 31 bulk, 29, 3Ot, 31,259-313, 364-371 cross-polarization coupling in, 344-346 optical processes in, 189-198 photorefractivity in, 319-401 use in optoelectronics, 218-226 Shadow masking, 83-84 Shockley-Read-Hall recombination, 10 Signal processing, coherent, 385, 385f. 386 Simplified band transport model, 323-324

408

INDEX

Single-beam transmittance measurements, 294 -29 5 Sommerfeld enhanced continuum, 182 Space-charge grating linear electro-optic effect, 326-327 plane-wave interference model, 321-323 simplified band transport model, 323324 steady-state solution to, 324-325 Spectroscopy, 218, 224f Spin-orbit split-off band, 179 Stark effects. See AC Stark effect; quadratic optical Stark effect (QSE); quantum confined Stark effect (QCSE) State filling. 3-6, 3f electrically controlled, 122- I23 enhanced nonlinearities based on, 64-75 Steady-state solution, to photoreactive effect, 324-325 Strain on coupled quantum wells, 113 on quantum wells, 108- 1 I3 Subbands, 18 Symmetric quantum wells, 103- 104

on zero-field exciton resonance, 125 Transmittance measurements, 294-295 Two-band model, of bound-electronic nonlinearities, 271-285 Two-parabolic-band model, 272, 276 Two-particle correlation effects, 206-219 Two-photon absorption (2PA). 271, 274275

U Urbach parameter, 7 Urbach tail. 7

v VanderLugt optical correlator, 388, 388f W

Wannier Stark localization (WSL), 107 Waveguides, 126 Weak-wave retardation, 260 Z

T Temperature-intensity resonance, 352-354, 363t Tensile strain, effect on QWs, 109- 110 Three-level system (3LS) model, 219 Transition rates, 273 Transmission, Fabry-Perot in, 138- 139 Transmission experiments, 123- 124 at long wavelengths. 124- 125

2-scan, 300-302, 304f excite-probe, 302-303, 305 ZnS, 371 nonlinearities in, 307t ZnSe, 371 nonlinearities in, 307t optical limiting data for, 310, 310f ZnTe, Characteristics of, 367, 368t

Contents of Volumes in This Series

Volume 1 Physics of 111-V Compounds C. Hilsurn, Some Key Features of 111-V Compounds Franc0 Bassuni, Methods of Band Calculations Applicable to 111-V Compounds E. 0. Kane, The k-p Method K L. Bonch-Brueuich, Effect of Heavy Doping on the Semiconductor Band Structure DonuId Long, Energy Band Structures of Mixed Crystals of 111-V Compounds Laura M. Roth and Petros N. Argyres, Magnetic Quantum Effects S. M. Puri and T. H. Geballe, Thermomagnetic Effects in the Quantum Region W. M. Becker, Band Characteristics near Principal Minima from Magnetoresistance E. H. Putley, Freeze-Out Effects, Hot Electron Effects, and Submillimeter Photoconductivity in InSb H. Weiss, Magnetoresistance Betsy Ancker-Johnson, Plasma in Semiconductorsand Semimetals

Volume 2 Physics of 111-V Compounds M. G. Holland, Thermal Conductivity S. I. Novkova, Thermal Expansion U.Piesbergen, Heat Capacity and Debye Temperatures G. Giesecke, Lattice Constants J. R Drabble, Elastic Properties A. U. Mac Rue and G. W. Gobeli, Low Energy Electron Diffraction Studies Robert Lee Mieher, Nuclear Magnetic Resonance Bernurd Goldstein, Electron Paramagnetic Resonance T. S. Moss,Photoconduction in 111-V Compounds E. Antoncik ad J. Tauc, Quantum Efficiency of the Internal Photoelectric EfTect in InSb G. W. Gobeli and I. G.Allen, Photoelectric Threshold and Work Function P. S. Pershan, Nonlinear Optics in 111-V Compounds M. Gershenzon, Radiative Recombination in the 111-V Compounds Frank Stern, Stimulated Emission in Semiconductors

409

410

CONTENTS OF VOLUMESIN THISSERIFS

Volume 3 Optical of Properties III-V Compounds Marvin Hass, Lattice Reflection Willium G. Spitzer, Multiphonon Lattice Absorption D. L Stierwalt and R F. Potter, Emittance Studies H. R Philipp and H. Ehrenveich, Ultraviolet Optical Properties Manuel Cardona. Optical Absorption above the Fundamental Edge Earnest J. Johnson. Absorption near the Fundamental Edge John 0. Dimmock, Introduction to the Theory of Exciton States in Semiconductors B. La.r and J. G. Mavroides, lnterband Magnetooptical Effects H. Y. Fan, Effects of Free Carries on Optical Properties Edward D. Palik and George E. Wright, Free-Carrier Magnetooptical Effects Richard If. Bube, Photoelectronic Analysis B. 0. Seruphin and H. E. Benneff,Optical Constants

Volume 4 Physics of I I - V Compounds N. A. Gorjwnova. A. S. Borschevskii, and D. N. Tretiukov, Hardness N. N. Sirota, Heats of Formation and Temperatures and Heats of Fusion of Compounds A”’BV

Don L. Kendull, Diffusion A. G. Chynoweth, Charge Multiplication Phenomena Robert W.Kqves, The Effects of Hydrostatic Pressure on the Properties of HI-V Semiconductors 1.. W. Aukerman, Radiation Effects N . A. Goryunova, F. P. Kesuman[v. and D.N. Nasledov, Phenomena in Solid Solutions R. T. Bate. Electrical Properties of Nonuniform Crystals

Volume 5 Infrared Detectors Henry Levinstein, Characterization of Infrared Detectors Puul W. Kruse, Indium Antimonide Photoconductive and PhotoelectromagneticDetectors M. B. Prince. Narrowband Self-Filtering Detectors fvurs Melngalis and T. C. Harman, Single-Crystal Lead-Tin Chalcogenides Donuld Long and Juseph L. Schmidt, Mercury-Cadmium Telluride and Closely Related Alloys E H. Putley, The Pyroelectric Detector Norman B. Stevens, Radiation Thermopiles R J. Kqves ond T. M. Quist, Low Level Coherent and Incoherent Detection in the Infrared M. C.Teich, Coherent Detection in the Infrared F R A r m . E. W. Surd, B. J. Peyton. und F. P. Pace, Infrared Heterodyne Detection with Gigahertz IF Response If S. Summers, Jr., Macrowave-Based Photoconductive Detector Roherr Sehr and Ruiner Zuleeg, Imaging and Display

Volume 6 Injection Phenomena hfurruy A. Lampert and Ronald B. Schilling, Current Injection in Solids: The Regional

Approximation Method Richard Williums, Injection by Internal Photoemission Allen M. Burnerr, Current Filament Formation

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R Baron and J. W. Mayer, Double Injection in Semiconductors W. Ruppel, The Photoconductor-Metal Contact

Volume 7 Application and Devices Part A John A. Copeland and Stephen Knight, Applications Utilizing Bulk Negative Resistance F. A. Padovani, The Voltage-Current Characteristics of Metal-Semiconductor Contacts P. L. Hower. W. W. Hooper. B. R Cairns, R D. Fairmun, and D. A. Tremere, The GaAs Field-Effect Transistor Marvin H. White, MOS Transistors G. R Antell, Gallium Arsenide Transistors T. L Tansley, Heterojunction Properties

Part B T. Misawa, IMPATT Diodes H. C. Okean, Tunnel Diodes Robert B. Campbell and Hung-Chi Chang, Silicon Junction Carbide Devices R E. Enstrom, H. Kressel, and L Krassner, High-Temperature Power Rectifiers of GaAs,-,P,

Volume 8 Transport and Optical Phenomena Richard J. Srirn, Band Structure and Galvanornagnetic Effects in 111-V Compounds with Indirect Band Gaps Roland W. Ure. Jr., Thermoelectric Effects in 111-V Compounds Herbert Piller, Faraday Rotation H. Barry Bebb and E W. William, PhotoluminescenceI: Theory E. W.Williams and H. Barry Bebb, Photoluminescence11: Gallium Arsenide

Volume 9 Modulation Techniques B. 0. Seraphin, Electroreflectance R L. Aggarwaf, Modulated Interband Magnetooptin Daniel F. Blossey and Paul Handler, Electroabsorption Bruno Bafz, Thermal and Wavelength Modulation Spectroscopy fvar BaMev, Piaopptical Effects D. E. Aspnes and N. Bortka, Electric-Field Effects on the Dielectric Function of Semiconductors and Insulators

Volume 10 Transport Phenomena R L. Rhode, Low-Field Electron Transport J, D. Wiley, Mobility of Holes in 111-V Compounds C. M. Wove and G. E. Stillman, Apparent Mobility Enhancement in Inhomogeneous Crystals Robert L Petersen, The Magnetophonon Effect

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Volume 11 Solar Cells Harold J. Hovel, Introduction; Carrier Collection, Spectral Response, and Photocurrent; Solar Cell Electrical Characteristics; Efficiency; Thickness; Other Solar Cell Devices; Radiation Effects Temperature and Intensity; Solar Cell Technology

Volume 12 Infrared Detectors (II) W. L. Eiseman. J. D. Merriam, and R F. Potter, Operational Characteristics of Infrared

Photodetectors Peter R Brutt, Impurity Germanium and Silicon Infrared Detectors E. H . Putley, lnSb Submillimeter Photoconductive Detectors G. E Stillman, C. M. Wove, and J. 0. Dimmock, Far-Infrared Photoconductivity in High Purity GaAs G. E. Stilhan und C. M. Wove, Avalanche Photodiodes P. L. Richards, The Josephson Junction as a Detector of Microwave and Far-Infrared Radiation E. H. Putley, The Pyroelectric Detector-An Update

Volume 13 Cadmium Telluride Kenneth Zanio, Materials Preparations; Physics; Defects; Applications

Volume 14 Lasers, Junctions, Transport N . Holonyak, Jr. und M. H. Lee, Photopumped Ill-V Semiconductor Lasers Henry Kressel and Jerome K Butler, Heterojunction Laser Diodes A Van dpr Ziel, Space-Charge-LimitedSolid-state Diodes Peter J. Price, Monte Carlo Calculation of Electron Transport in Solids

Volume 15 Contacts, Junctions, Emitters B. L. Shurma, Ohmic Contacts to Ill-V Compounds Semiconductors Allen Nussbaum, The Theory of Semiconducting Junctions

John S. Escher, NEA Semiconductor Photoemitters

Volume 16 Defects, (HgCd)Se, (HgCd)Te Henry Kressel, The Erect of Crystal Defects on Optoelectronic Devices C: R Whitsett. J. G. Broerman. and C. J. Summers, Crystal Growth and Properties of Hg, ,Cd,Se alloys 61. H. Weiler, Magnetooptical Properties of H& -,Cd,Te Alloys Puul W Kruse and John G. Ready, Nonlinear Optical Effects in Hg-,Cd,Te

Volume 17 CW Processing of Silicon and Other Semiconductors Jumes F. Gibbons. Beam Processing of Silicon Arlo Lietoilu. Richard B. Gold, Jumes F. Gibbons, and Lee A . Christef,Temperature Distribu-

CONTENTS OF VOLUMESIN THISSERIES

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tions and Solid Phase Reaction Rates Produced by Scanning CW Beams Arto Leitoila and James F. Gibbons, Applications of CW Beam Processing to Ion Implanted Crystalline Silicon N. M.Johnson, Electronic Defects in CW Transient Thermal Processed Silicon K F. Lee, T. J. Stultz, and James F. Gibbons, Beam Recrystallized Polycrystalline Silicon: Properties, Applications,and Techniques T. Shibata, A. Wakita. T. W. Sigmon, and James F. Gibbons, Metal-Silicon Reactions and Silicide Yves I. Nissim and James F. Gibbons, CW Beam Processing of Gallium Arsenide

Volume 18 Mercury Cadmium Telluride Paul W.Kruse, The Emergence of (H&-,Cd,)Te as a Modem Infrared Sensitive Material H. E Hirsch, S. C. Liang. and A. G. White, Preparation of High-Purity Cadmium, Mercury, and Tellurium W. F. H. Micklethwaire, The Crystal Growth of Cadmium Mercury Telluride Paul E. Petersen, Auger Recombination in Mercury Cadmium Telluride R M. Broudy and V. J. Mazurczyck, (HgCd)Te Photoconductive Detectors M.B. Reine, A, K Soad, and T. J. Tredwell, Photovoltaic Infrared Detectors M.A. Kinch, Metal-Insulator-Semiconductor Infrared Detectors

Volume 19 Deep Levels, GaAs, Alloys, Photochemistry G. F. Neumark and K Kosai, Deep Levels in Wide Band-Gap 111-V Semiconductors David C. Look, The Electrical and Photoelectronic Properties of Semi-InsulatingGaAs R F. Brebrick, Ching-Hua Su. and P o k - h i Liao, Associated Solution Model for Cia-In-Sb and Hg-Cd-Te Yu. Y a Gurevich and Yu. V. Pleskon, Photoelectrochemistry of Semiconductors

Volume 20 Semi-InsulatingGaAs R N. Thomas. H. M. Hobgood, G. W.Eldridge, D. L Barrett, T. T.Braggins. L. B. Ta, and S. K Wang, High-Purity LEC Growth and Direct Implantation of GaAs for Monolithic Microwave Circuits C. A. Srolte, Ion Implantation and Materials for GaAs Integrated Circuits C. G. Kirkpatrick, R T. Chen, D. E. Holmes. P. M.Asbeck, K R Elliort, R D. Fairman, and J. R Oliver, LEC GaAs for Integrated Circuit Applications J. S. Blakemore and S. Rahimi, Models for Mid-Gap Centers in Gallium Arsenide

Volume 21 Hydrogenated Amorphous Silicon Part A Jacques I. Pankove, Introduction Masataka Hirose, Glow Discharge; Chemical Vapor Deposition Yoshiyuki Uchida, di Glow Discharge T. D. Moustakas, Sputtering Isao Yamada, Ionized-Cluster Beam Deposition Bruce A. Scott, Homogeneous Chemical Vapor Deposition

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CONTENTS OF VOLUMESIN THISSERIES

Frank J. Kumpas, Chemical Reactions in Plasma Deposition Paul A. Longewuy, Plasma Kinetics Herbert A . Weukliem, Diagnostics of Silane Glow Discharges Using Probes and Mass Spectroscopy Lester Glurtmn, Relation between the Atomic and the Electronic Structures A. Chenevur-Paule, Experiment Determination of Structure S. Minomura, Pressure Effects on the Local Atomic Structure Duvid Adkr, Defects and Density of Localized States

Part B Jucques 1. Punkove, Introduction C. D. Codv, The Optical Absorption Edge of a-Si: H Nubil M. Amer and Warren 5. Jackson, Optical Properties of Defect States in a-Si: H P. J. Zunzucchi, The Vibrational Spectra of a-Si: H Yoshihiro Humakawa, Electroreflectanceand Electroabsorption Jefrey S. Lannin, Raman Scattering of Amorphous Si, Ge, and Their Alloys R A . Street, Luminescence in a-Sk H Richard S. Crundull, Photoconductivity J. Tam, Time-Resolved Spectroscopy of Electronic Relaxation Processes P. B Vunier, IR-Induced Quenching and Enhancement of Photoconductivity and Photoluminescence H. Schude, Irradiation-Induced Metastable Effects L. Ley. Photoelectron Emission Studies

Part C Jucques I. Punkow, Introduction J David Cohen. Density of States from Junction Measurements in Hydrogenated Amorphous Silicon P. C. Tuylur, Magnetic Resonance Measurements in a-Si: H K Moriguki, Optically Detected Magnetic Resonance J Dresner, Carrier Mobility in a-Si H T. Tie&, Information about band-Tail States from Time-of-Flight Experiments Arnold R Moore, Diffusion Length in Undoped a-Si: H W. Beyer and J. OverhoJ Doping Effects in a-Si: H II. Frirzche, Electronic Properties of Surfaces in a-Si:H C. R Wronski, The Staebler-Wronski Effect R J. Nemunich, Schottky Barriers on a-Si: H B. Ahetes und T. Tiedje. Amorphous Semiconductor Superlattices

Part D Jacques I. Punko ve, In trod uct ion D. E. Curtson, Solar Cells G. A . Swurr;, Closed-Form Solution of I-V Characteristic for a a-Si: H Solar Cells bumu Shimizu, Elect rophotography Smhio Ishioku, Image Pickup Tubes

CONTENTS OF VOLUMESIN THISSERIES

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P. G. LeComber and W.E. Spear, The Development of the a-Sk H Field-Effect Transistor and Its Possible Applications D. G. Ast, a-Si: H FET-Addressed LCD Panel S. Kaneko, Solid-state Image Sensor Masakiyo Matsumura, Charge-Coupled Devices M. A. Bosch, Optical Recording A. D'Amico and G. Fortunato, Ambient Sensors Hiroshi Kukimoto, Amorphous Light-Emitting Devices Robert J. Phelan, Jr., Fast Detectors and Modulators Jacques I. Pankove, Hybrid Structures P. G. LeComber, A. E. Owen. W. E. Spear, J. Hajto, and W. K Choi, Electronic Switching in Amorphous Silicon Junction Devices

Volume 22 Lightwave Communications Technology Part A Kazuo Nakajima, The Liquid-Phase Epitaxial Growth of IngaAsp W. T. Tsang, Molecular Beam Epitaxy for Ill-V Compound Semiconductors G. B. Stringfellow, Organometallic Vapor-Phase Epitaxial Growth of Ill-V Semiconductors G. Beuchet, Halide and Chloride Transport Vapor-Phase Deposition of InGaAsP and GaAs Manijeh Razeghi, Low-Pressure Metallo-Organic Chemical Vapor Deposition of Ga,in,-,AsP,-, Alloys P. M. PetroJ Defects in 111-V Compound Semiconductors

Part B J. P. van der Ziel, Mode Locking of Semiconductor Lasers Kam Y. Lau and Ammon Yariv, High-Frequency Current Modulation of Semiconductor Injection Lasers Charles H. Henry, Special Properties of Semiconductor Lasers Yasuharu Suematsu. Katswni Kishino, Shigehisa Arai, and Fumio Koyama. Dynamic SingleMode Semiconductor Lasers with a Distributed Reflector W. T. Tsang, The Cleaved-Coupled-Cavity(C3) Laser

Part C R J. Nelson and N. K Dutta, Review of InGaAsP 1nP Laser Structures and Comparison of

Their Performance N. Chinone and M. Nakamura, Mode-Stabilized Semiconductor Lasers for 0.7-0.8- and 1.1 - I .6-pm Regions Yoshiji Horikoshi, Semiconductor Lasers with Wavelengths Exceeding 2prn B. A. Dean and M. Dixon, The Functional Reliability of Semiconductor Lasers as Optical Transmitters R H. Saul, T. P. Lee, and C. A. Burus, Light-Emitting Device Design C. L Zipfel, Light-Emitting Diode-Reliability Tien Pei Lee and Tingye Li, LED-Based Multimode Lightwave Systems Kinichiro Ogawa, Semiconductor Noise-Mode Partition Noise

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CONTENTS OF VOLUMES IN THIS SERIEs

Part D Fea'erico Capasso, The Physics of Avalanche Photodiodes T. P. Pearsall and M. A. Pollack, Compound Semiconductor Photodiodes Taka0 Kaneda, Silicon and Germanium Avalanche Photodiodes S. R Forrest, Sensitivity of Avalanche Photodetector Receivers for High-Bit-Rate LongWavelength Optical Communication Systems J. C. Campbell, Phototransistors for Lightwave Communications

Part E Shyh Wang, Principles and Characteristics of Integrable Active and Passive Optical Devices Shlomo Margalit and Amnon Yariv, Integrated Electronic and Photonic Devices T'aoki Mukai, Yoshihisa Y m m r o . and Tatsuya Kimura, Optical Amplification by Semiconductor Lasers

Volume 23 Pulsed Laser Processing of Semiconductors R E Wood, C. W. Whire,and R T Young, Laser Processing of Semiconductors: An Overview C'. W. White, Segregation, Solute Trapping, and Supersaturated Alloys G. E Jellison, Jr., Optical and Electrical Properties of Pulsed Laser-Annealed Silicon R F. Wood ond G. E. Jellison, Jr., Melting Model of Pulsed Laser Processing R. F. Wood and F. W. Young, Jr., Nonequilibrium Solidification Following Pulsed Laser Melting

D. H.

Lowndes and C. E. Jellison. Jr., Time-Resolved Measurement During Pulsed Laser irradiation of Silicon D. M. Zebner, Surface Studies of Pulsed Laser Irradiated Semiconductors D. H. Lowndes, Pulsed Beam Processing of Gallium Arsenide R B. James, Pulsed CO, Laser Annealing of Semiconductors R T. Young and R E Wood, Applications of Pulsed Laser Processing

Volume 24 Applications of Muhiquantum Wells, Selective Doping, and Superlattices C. Weisbuch, Fundamental Properties of 111-V Semiconductor Two-Dimensional Quantized Structures: The Basis for Optical and Electronic Device Applications If. Morkoc und H. Unlu, Factors Affecting the Performance of (Al,Ga)As/GaAs and (Al,Ga)As/inGaAs Modulation-Doped Field-Effect Transistors: Microwave and Digital Applications N . T. Linh, Two-Dimensional Electron Gas FETs: Microwave Applications M. Ahe et ul,, Ultra-High-speed HEMT Integrated Circuits D. S. Chemla, D. A. 8. Miller, and P. W. Smith, Nonlinear Optical Properties of Multiple Quantum Well Structures for Optical Signal Processing F. Capasso, Graded-Gap and Superlattice Devices by Band-Gap Engineering iV T. Tsang, Quantum Confinement Heterostructure Semiconductor Lasers G. C. Osbourn el ul., Principles and Applications of Semiconductor Strained-Layer SuperlattiCeS

CONTENTS OF VOLUMESIN THISSERIES

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Volume 25 Diluted Magnetic Semiconductors W. Giriat and J. K Furdyna, Crystal Structure, Composition, and Materials Preparation of Diluted Magnetic Semiconductors W. M. Becker, Band Structure and Optical Properties of Wide-Gap A/'_,Mn,B'" Alloys at Zero Magnetic Field Saul Oserof and Pieter H. Keesom, Magnetic Properties: Macroscopic Studies Giebultowicz and T. M. Holden, Neutron Scattering Studies of the Magnetic Structure and Dynamics of Diluted Magnetic Semiconductors J. Kossut, Band Structure and Quantum Transport Phenomena in Narrow-Gap Diluted Magnetic Semiconductors C. Riquaux, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. A. Gaj, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. Mycielski, Shallow Acceptors in Diluted Magnetic Semiconductors:Splitting, Boil-off, Giant Negative Magnetoresistance A. K. R a m a h and R Rodriquez, Raman Scattering in Diluted Magnetic Semiconductors P. A. WolK Theory of Bound Magnetic Polarons in Semimagnetic Semiconductors

Volume 26 I I L V Compound Semiconductors and Semiconductor Properties of Superionic Materials Zou Yuanxi, 111-V Compounds H. Y. Winston, A. T. Hunter, H. Kimura, and R E. Lee, InAs-Alloyed GaAs Substrates for Direct Implantation P. K. Bhattachary andS. Dhar, Deep Levels in Ill-V Compound SemiconductorsGrownby MBE Yu. Yu Gurevich and A. K. Ivanov-Shits, Semiconductor Properties of Supersonic Materials

Volume 27 High Conducting Quasi-One-DimensionalOrganic Crystals E. M. Conwell, Introduction to Highly Conducting Quasi-One-DimensionalOrganic Crystals I. A. Howard, A Reference Guide to the Conducting Quasi-OnaDimensional Organic Molecular Crystals J. P. Pouquet, Structural Instabilities E. M. Conwell, Transport Properties C. S. Jacobsen, Optical Properties J. C. Scott, Magnetic Properties L. Zuppiroli, Irradiation Effects: Perfect Crystals and Real Crystals

Volume 28 Measurement of High-speed Signals in Solid State Devices J. Frey and D. loannou, Materials and Devices for High-speed and OptoelectronicApplications H. Schumacher and E. Strid, Electronic Wafer Probing Techniques D. H. Auston, Picosecond Photoconductivity: High-speed Measurements of Devices and Materials J. A. V a l h n i s , Electro-Optic Measurement Techniques for Picosecond Materials, Devices, and Integrated Circuits. J. M. Wiesenfeldand R K Jain, Direct Optical Probing of Integrated Circuits and High-speed Devices G. Plows, Electron-Beam Probing A . M. Weiner and R B. Marcus, Photoemissive Probing

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CONTENTS OF VOLUMFSIN THISSERIES

Volume 29 Very High Speed Integrated Circuits: Gallium Arsenide LSI M. Kuzuhara and T. Nmaki, Active Layer Formation by Ion Implantation H. Hasimoro, Focused Ion Beam Implantation Technology T. Nozaki and A. Higushisaka, Device Fabrication Process Technology M. In0 und T. Tukada, GaAs LSI Circuit Design M ,Hira.vumu, M. Ohmori. and K Yamsuki, GaAs LSI Fabrication and Performance

Volume 30 Very High Speed Integrated Circuits: Heterostructure H. Warmube. T. Mizutani, and A. Usui, Fundamentals of Epitaxial Growth and Atomic Layer

Epitaxy S. Hiyamizu, Characteristics of Two-Dimensional Electron Gas in 111-V Compound Heterostructures Grown by MBE T. Nakanisi, Metalorganic Vapor Phase Epitaxy for High-Quality Active Layers 7.Nimura, High Electron Mobility Transistor and LSI Applications T. Sugera and T. Ishibushi, Hetero-Bipolar Transistor and LSI Application H . Matsueda. T. Tanuka, and M. N a k m r a , Optcelectronic Integrated Circuits

Volume 31 Indium Phospbide: Crystal Growth and Characterization J. f . Forges, Growth of Discoloration-free InP M.J. McCollum and C. E. Stillman, High Purity InP Grown by Hydride Vapor Phase Epitaxy T. /nuah und T. F u k h , Direct Synthesis and Growth of Indium Phosphide by the Liquid Phosphorous Encapsulated Czochralski Method 0. O h , K. Katagiri. K Shinohara. S. Katsura, Y. Takahashi. K Kainosho. K Kohiro. and R Hiruno, InP Crystal Growth, Substrate Preparation and Evaluation K Tuda, M. Tatsumi. M. Morioka, T. Araki. and T. Kawuse, InP Substrates: Production and Quality Control M. Razeghi, LP-MOCVD Growth, Characterization, and Application of InP Material T A. Kenne4v and P. J. Lin-Chung, Stoichiometric Defects in InP

Volme 32 Strained-Layer Superlattices Physics T. f . f e u r s d . Strained-Layer Superlattices Fred H . Pollack, Effects of Homogeneous Strain on the Electronic and Vibrational Levels in Semiconductors J. Y. Marzin, J. M. Gerurd, f . Voisin. and J. A. Erum, Optical Studies of Strained 111-V Heterolayers R. People and S. A. Jackson, Structurally Induced States from Strain and Confinement M. Juros. Microscopic Phenomena in Ordered Suprlattices

Volume 33 Strained-Layer Superlattices: Materials Sience and Technology R. Hull and J. C. &an, Principles and Concepts of Strained-Layer Epitaxy

William J. Schulp; Paul J. Tusker, Marc C. Foisy. and Lester I? Eustman, Device Applications of Strained-Layer Epitaxy

CONTENTS OF VOLUMES IN THISSERIES

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S. 7: Picraux, B. L. Doyle, and J. I: Tsao, Structure and Characterization of Strained-Layer

Superlattices E. Kasper and F. Schafer, Group IV Compounds Dale L. Martin, Molecular Beam Epitaxy of IV-VI Compounds Heterojunction Robert L. Gunshor. Leslie A. Kolodziejski, Arto V. Nurmikko, and Nobuo Otsuka, Molecular Beam Epitaxy of 11-VI Semiconductor Microstructures

Volume 34 Hydrogen in Semiconductors J. I. Pankove and N. M. Johnson, Introduction to Hydrogen in Semiconductors C. H. Seager, Hydrogenation Methods J. I. Pankove, Hydrogenation of Defects in Crystalline Silicon J. W. Corbett, P. Derik, U.!k Desnica, and S. J . Pearton, Hydrogen Passivation of Damage Centers in Semiconductors S. J. Pearton, Neutralization of Deep Levels in Silicon J. I. Pankove, Neutralization of Shallow Acceptors in Silicon N. M. Johnson, Neutralization of Donor Dopants and Formation of Hydrogen-Induced Defects in n-Type Silicon M. Stavola and S. J. Pearton, Vibrational Spectroscopy of Hydrogen-Related Defects in Silicon A. D. Marwick, Hydrogen in Semiconductors: Ion Beam Techniques C. Herring and N. M. Johnson, Hydrogen Migration and Solubility in Silicon E. E. Huller, Hydrogen-Related Phenomena in Crystalline Germanium J. Kakalios, Hydrogen Diffusion in Amorphous Silicon J. Chevalier, B. Clerjauud, and B. Pajot, Neutralization of Defects and Dopants in 111-V Semiconductors G. G.DeLeo and W. B. Fowler, Computational Studies of Hydrogen-Containing Complexes in Semiconductors R F. Kiej and T. L. Estle, Muonium in Semiconductors C. G. Van de Walle, Theory of Isolated Interstitial Hydrogen and Muonium in Crystalline Semiconductors

Volume 35 Nanostructured Systems Mark Reed, Introduction H. van Houten, C. W. J. Beenakker, and B. J. van Wees, Quantum Point Contacts G. Timp, When Does a Wire Become an Electron Waveguide? M. Buttiker, The Quantum Hall Effects in Open Conductors W. Hansen. J. P. Kotthaus, and U. Merkt, Electrons in Laterally Periodic Nanostructures

Volume 36 The Spectroscopy of Semiconductors D. Heimun, Spectroscopyof Semiconductors at Low Temperatures and High Magnetic Fields Arto K Nurmikko, Transient Spectroscopy by Ultrashort Laser Pulse Techniques A. L Ramdas and S. Rodriguez, Piezospectroscopy of Semiconductors Orest J. Glembocki and Benjamin V. Shanabrook, Photoreflectance Spectroscopyof Microstructures David G. Seiler, Christopher L. Littler, and Margaret H. Wiler, One- and Two-Photon Magneto-Optical Spectroscopy of InSb and Hg, -,Cd,Te

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Volume 37 The Mechnical Properties of Semiconductors 4.4. Chen. Arden Sher and W. T. Yost, Elastic Constants and Related Properties of Semiconductor Compounds and Their Alloys David R Clarke, Fracture of Silicon and Other Semiconductors Hans SiPrholP; The Plasticity of Elemental and Compound Semiconductors Sivaruman Guruswmy, Katherine T. Faber and John P. Hirrh, Mechanical Behavior of Compound Semiconductors Suhhanh Mohajan, Deformation Behavior of Compound Semiconductors John P. Hirth, Injection of Dislocations into Strained Multilayer Structures Don Kendall, Charles B. Fle&rmunn, and Kevin J. Malloy, Critical Technologies for the Micromachining of Silicon Ikuo Mutsuba and Kinji Mokuya, Processing and Semiconductor Thennoelastic Behavior

Volume 38 Imperfections in IIW Materials lido Scherz and Marrhiar S c k j k r , Density-Functional Theory of spBonded Defects in W/V Semiconductors Maria Kuminska and Eicke R Weber, El2 Defect in GaAs Duoid C. bok, Defects Relevant for Compensation in Semi-Insulating GaAs R C. Newmun, Local Vibrational Mode Spectroscopy of Defects in III/V Compounds Andrzej M. Hennel, Transition Metals in III/V Compounds Kevin J. Malloy and Ken Khuchaturyan, DX and Related Defects in Semiconductors Y. Swmino/han and Andrew S. Jordan, Dislocations in HI/V Compounds Krzysztoj W. Nauka, Deep Level Defects in the Epitaxial HI/%’ Materials

Volume 39 Minority Carriers in 111-V Semiconductors Physics and Applications Niloy K Duttu, Radiative Transitions in GaAs and Other 111-V Compounds Richard K Ahrenkiel, Minority-Carrier Lifetime in 111-V Semiconductors Tomofumi FUIWIU,High Field Minority Electron Transport in pGaAs Murk S. Lundsfrom, Minority-Carder Transpon in 111-V Semiconductors Richard A. Abram. Effects of Heavy Doping and High Excitation on the Band Structure of GaAs David Yevick and Witold Bardyszewski, An Introduction to Non-Equilibrium Many-Body Analyses of Optical Processes in 111-V Semiconductors

Volume 40 Epitaxial Microstrwtures E. F. Schuberr, Delta-Doping of Semiconductors: Electronic, Optical, and Structural Properties of Materials and Devices A Gossard, M.Sundarum. and P. Hopkins, Wide Graded Potential Wells P. PetroK Direct Growth of Nanometer-Size Quantum Wire Superlattices E. Kapon, Lateral Patterning of Quantum Well Heterostructures by Growth of Nonplanar Subst rates H. Temkin. D. Gershoni, and M. Punish, Optical Properties of Gal-,ln,As/lnP Quantum Wells

CONTENTS OF VOLUMES INTHISSERE

42 1

Volume 41 High Speed Heterostructure Devices F. Capasso, F. Beltram, S. Sen, A. Pahlevi, and A. Y. Cho, Quantum Electron Devices: Physics and Applications P. Solomon, D. J. Frank, S.L. Wright, and F. Canora, GaAs-Gate Semiconductor-lnsulatorSemiconductor FET M. H. Hashemi and U.K Mishra, Unipolar InP-Based Transistors R Kiehl, Complementary Heterostructure FET Integrated Circuits T.Ishibashi, GaAs-Based and InP-Based Heterostructure Bipolar Transistors H. C. Liu and T. C. L G. Sollner, High-Frequency-TunnelingDevices H. Ohnishi, T. More, M. Takatsu, K Imamura, and N. Yokoyama, Resonant-Tunneling Hot-Electron Transistors and Circuits

Volume 42 Oxygen in Silicon F. Shimura, Introduction to Oxygen in Silicon W. Lin, The Incorporation of Oxygen into Silicon Crystals T. J. Schaffher and D. K Schroder, CharacterizationTechniques for Oxygen in Silicon W. M. Bullis, Oxygen Concentration Measurement S.M. Hu, Intrinsic Point Defects in Silicon B. Pajot, Some Atomic Configurations of Oxygen J. Michel and L. C. Kimerling, Electical Properties of Oxygen in Silicon R C. Newman and R Jones, Diffusion of Oxygen in Silicon T. Y. Tan and W.J. Taylor, Mechanisms of Oxygen Precipitation: Some Quantitative Aspects M. Schrems, Simulation of Oxygen Precipitation K Simino and I. Yonenaga, Oxygen Effect on Mechanical Properties W. Bergholz, Grown-in and Process-Induced Effects F. Shimura, Intrinsic/Internal Gettering H. Tsuya, Oxygen Effect on Electronic Device Performance

Volume 43 Semiconductors for Room Temperature Nuclear Detector Applications R B. James and T.E Schlesinger, Introduction and Overview L. S. Darken and C. E Cox, High-Purity Germanium Detectors A. Burger, D. Nason. L Van den Berg, and M. Schieber, Growth of Mercuric Iodide X. J. Bao, T. E. Schlesinger, and R B. James, Electrical Properties of Mercuric Iodide X. J. Bao, R B. James, and T.E Schlesinger, Optical Properties of Red Mercuric Iodide M.Hage-Ali and P. Sirerr, Growth Methods of CdTe Nuclear Detector Materials M. Hage-Afi and P Siffert, Characterization of CdTe Nuclear Detector Materials M. Huge-Ali and P. S@ert, CdTe Nuclear Detectors and Applications R B. James, T. E Schlesinger, J. Lund. and M. Schieber, Cd, -xZnxTe Spectrometers for Gamma and X-Ray Applications D. S. McGregor, J. E. f f i m r a a d , Gallium Arsenide Radiation Detectors and Spectrometers J. C. Lund, I? Olschner. and A. Burger, Lead Iodide M. R Squillante. and K S. Shah, Other Materials: Status and Prospects K M. Cerrish, Characterization and Quantification of Detector Performance J. S. fwanczyk and B. E Putt, Electronics for X-ray and Gamma Ray Spectrometers M. Schieber, R B. James, and T.E. Schlesinger, Summary and Remaining Issues for Room Temperature Radiation Spectrometers

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Volume 44 II-Iv BlwlGreen Light Emitters: Device Physics and Epitaxial Growth J. Hun and R L. Gunshor, MBE Growth and Electrical Properties of Wide Bandgap ZnSe-based II-VI Semiconductors Shizuo Fujitu and Shigeo Fujitu, Growth and Characterization of ZnSe-based Il-VI Semiconductors by MOVPE Eusen Ho und Leslie A . Kolodziejski, Gaseous Source UHV Epitaxy Technologies for Wide Bandgap 11-VI Semiconductors Chris C. Yon de Wulle, Doping of Wide-Band-Gap 11-VI Compounds-Theory Roberto Cingoluni, Optical Properties of Excitons in ZnSe-Based Quantum Well Heterostructures A . ishibushi und A. V. Nurmikko, II-VI Diode Lasers: A Current View of Device Performance and Issues Suprurik Cuhu and John Petruzello, Defects and Degradation in Wide-Gap 11-VI-based Structures and Light Emitting Devices

Volume 45 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Electrical and Physiochemical Characterization Heiner Ryssel. Ion Implantation into Semiconductors: Historical Perspectives

You-Nian Wung and Teng-Cui Mu, Electronic Stopping Power for Energetic Ions in Solids Suchiko T. Nukuguwu, Solid Effect on the Electronic Stopping of Crystalline Target and Application to Range Estimation G. MGller, S. Kulbitzer and G. N. Greaves. Ion Beams in Amorphous Semiconductor Research Jumunu Boussey-Said. Sheet and Spreading Resistance Analysis of Ion Implanted and Annealed Semiconductors M. L. Polignuno und G. Queirolo. Studies of the Stripping Hall Effect in Ion-Implanted Silicon J. Stoemenos, Transmission Electron Microscopy Analyses Roberta Nipoti und Murco Srrvidori, Rutherford Backscattering Studies of Ion Implanted Semiconductors P. Zuumseil. X-ray Diffraction Techniques

Volume 46 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Optical and Photothermal Characterization M. Fried, T. Lohner and J. Cyului, Ellipsometric Analysis Antonios Seus and Constuntinos Christojdes, Transmission and Reflection Spectroscopy on Ion Implanted Semiconductors Andreus Othonos and Constuntinos Christojides, Photoluminescence and Raman Scattering of Ion Implanted Semiconductors. Influence of Annealing (bnstuntinos Chrisrojides, Photomodulated Thermoreflectance Investigation of Implanted Wafers. Annealing Kinetics of Defects Lf. Zammit, Photothermal Deflection Spectroscopy Characterization of Ion-Implanted and Annealed Silicon Films Andreus Mundelis. Arief Budirnun und Miguel Vargus, Photothermal Deep-Level Transient Spectroscopy of Impurities and Defects in Semiconductors K. Kulish and S. Churbonneuu, Ion Implantation into Quantum-Well Structures ..tlr.uandre M. Myusnikov und Nikoluy N. Gerusimenko, Ion Implantation and Thermal Annealing of Ill-V Compound Semiconducting Systems: Some Problems of Ill-V Narrow Gap Semiconductors

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Volume 47 Uncooled Infrared Imaging Arrays and Systems R G. Buser and M. P. Tompsett, Historical Overview P . W Kruse, Principles of Uncooled Infrared Focal Plane Arrays R. A . Wood, Monolithic Silicon Microbolometer Arrays C. M. Hanson, Hybrid Pyroelectric-Ferroelectnc Bolometer Arrays D. L. Polla and J . R. Choi, Monolithic Pyroelectric Bolometer Arrays N . Teranishi, Thermoelectric Uncooled Infrared Focal Plane Arrays M. F. Tompsett, Pyroelectric Vidicon 7: W Kenny, Tunneling Infrared Sensors J . R. Vig, R. L. Filler and E Kim, Application of Quartz Microresonatorsto Uncooled Infrared Imaging Arrays P . W Kruse, Application of Uncooled Monolithic Thermoelectric Linear Arrays to Imaging Radiometers

Volume 48 High Brightness Light Emitting Diodes C. B. Stringfellow, Materials Issues in High-Brightness Light-Emitting Diodes M. G. Craford, Overview of Device issues in High-Brightness Light-Emitting Diodes F. M. Sreranku, AIGaAs Red Light Emitting Diodes C. H. Chen, S . A. Stockman, M . J . Peanasky, and C. P. Kuo, OMVPE Growth of AlGaInP for High Efficiency Visible Light-Emitting Diodes F. A . Kish and R. M . Fletcher, AlGaInP Light-Emitting Diodes M. W Hodapp, Applications for High Brightness Light-Emitting Diodes I. Akasaki and H . Amano, Organometallic Vapor Epitaxy of GaN for High Brightness Blue Light Emitting Diodes S . Nakamura, Group 111-V Nitride Based Ultraviolet-Blue-Green-Yellow Light-Emitting Diodes and Laser Diodes

Volume 49 Light Emission in Silicon: from Physics to Devices David J. Lockwood, Light Emission in Silicon Gerhurd Abstreiter, Band Gaps and Light Emission in Si/SiGe Atomic Layer Structures Thomas G. Brown and Dennis G. Hall, Radiative Isoelectronic Impurities in Silicon and Silicon-GermaniumAlloys and Superlattices J. Michel, L. K C.Assali. M. T. Morse. and L. C. Kimerling. Erbium in Silicon Yoshihiko Kanemitsu, Silicon and Germanium Nanoparticles Phirippe M. Faucher,Porous Silicon: Photoluminescenceand Electroluminescent Devices C. Delerue. G. Allan. and M. Lannoo. Theory of Radiative and Nonradiative Processes in Silicon Nanocrystallites Louis Brus, Silicon Polymers and Nanocrystals

Volume 50 Gallium Nitride (GaN) J. I. Pankove and T. D. Moustakas, Introduction S. P. DenBaars and S. Keller. Metalorganic Chemical Vapor Deposition (MOCVD) of Group Ill Nitrides W. A. Bryden and T. J. Kistenmacher, Growth of Group 111-A Nitrides by Reactive Sputtering N. Newman, Thennochemistry of Ill-N Semiconductors S. J. Pearton and R J. Shul, Etching of Ill Nitrides

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S. M. Bedair, Indium-based Nitride Compounds A. Trampert. 0. Brandt, and K H. PIoog, Crystal Structure of Group Ill Nitrides

H. Morkoc, F. H m d a n i . and A. Salvador. Electronic and Optical Properties of Ill-V Nitride based Quantum Wells and Superlattices K Doverspike andJ. I. Punkowe. Doping in the 111-Nitrides T. Suski and P. Perlin. High Pressure Studies of Defects and Impurities in Gallium Nitride B. Monemur. Optical Properties of GaN W. R L. Lombrecht. Band Structure of the Group 111 Nitrides N. E. Christensen and P. Perlin, Phonons and Phase Transitions in GaN S. Nakamura, Applications of LEDs and LDs I. Akasaki and ff. Amano, Lasers J. A. Cooper, Jr.. Nonvolatile Random Access Memories in Wide Bandgap Semiconductors

Volume 51A Identification of Defects in Semiconductors George D. Watkins, EPR and ENDOR Studies of Defects in Semiconductors J.-M. Spaerh, Magneto-Optical and Electrical Detection of Paramagnetic Resonance in Semiconductors T. A. Kennedy and E. R Glaser, Magnetic Resonance of Epitaxial Layers Detected by Photoluminescence K H. Chow, B. Hitri. and R E Kiefl, pSR on Muonium in Semiconductorsand Its Relation to Hydrogen K M o Saarinen. Pekka Hautojdrvi, and Catherine Corhel, Positron Annihilation Spectroscopy of Defects in Semiconductors R Jones and P. R Briddon. The Ah Initio Cluster Method and the Dynamics of Defects in Semiconductors

Volume SIB Identification of Defects in Semiconductors Gordon Davies, Optical Measurements of Point Defects P. M. Mooney, Defect Identification Using Capacitance Spectroscopy Michael Stavola, Vibrational Spectroscopy of Light Element Impurities in Semiconductors P. Schwander. W. D. R m , C. Kisielowski, M. Gribelyuk, and A. Ourmazd Defect Processes in Semiconductors Studied at the Atomic Level by Transmission Electron Microscopy Nikos D. Jager and Eicke R Weber. Scanning Tunneling Microscopy of Defects in Semiconductors

Volume 52 SIC Materials and Devices Kenneth Jcirrendahl and Robert F. Davis, Materials Properties and Characterization of Sic V. A. Dmitriev and M. G. Spencer, Sic Fabrication Technology: Growth and Doping C: Saxena and A. J. Sreckl, Building Blocks for SIC Devices: Ohmic Contacts, Schottky Contacts, and pn Junctions Michael S. Shur, Sic Transistors C. D. Brandt, R C. Clarke, R R Siergiej. J. B. Casady, A. W. Morse. S. Sriram. and A. K Agurwal, Sic for Applications in High-Power Electronics R J . Tren, Sic Microwave Devices

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J. Edmond. H. Kong, G Negley, M.Leonard K Doverspike, K Weeks, A. Suvorov, D. Waltz, and C. Carter, Jr., Sic-Based UV Photodiodes and Light Emitting Diodes Hadis Morkoc, Beyond Silicon Carbide! 111-V Nitride-Based Heterostructures and Devices

Volume 53 Cumulative Subject and Author Index Including Tables of Contents for Volume 1-50 Volume 54 High Pressure in Semiconductor Physics I William Paul, High Pressure in Semiconductor Physics: A Historical Overview N. E. Christensen. Electronic Structure Calculations for Semiconductorsunder Pressure R J. Neirnes and M. I. McMahon, Structural Transitions in the Group IV, 111-V and 11-VI Semiconductors Under Pressure A. R Goni and K Syassen, Optical Properties of Semiconductors Under Pressure Pawel Trautmn, Michal Baj, and Jcek M. Baranowski Hydrostatic Pressure and Uniaxial Stress in Investigations of the EL2 Defect in GaAs Ming-fu Li and Peter Y. Yu, High-pressure Study of DX Centers Using Capacitance Techniques Tadeusz Suski, Spatial Correlations of Impurity Charges in Doped Semiconductors Noritaka Kuroda, Pressure Effects on the Electronic Properties of Diluted Magnetic Semiconductors

Volume 55 High Pressure in Semiconductor Physics Il D. K Maude and J. C. Porral, Parallel Transport in Low-Dimensional Semiconductor Structures P. C. Klipstein. Tunneling Under Pressure: High-pressure Studies of Vertical Transport in Semiconductor Heterostructures Evangelos Anastassakis and Manuel Cardona, Phonons, Strains, and Pressure in Semiconductors FredH Poll& EKects of External Uniaxial Stress on the Optical Properties of Semiconductors and Semiconductor Microstructures A. R Adams, M. Silver, and J. Allarn, Semiconductor Optoelectronic Devices S. Porowski and I. Grzegory, The Application of High Nitrogen Pressure in the Physics and Technology of 111-N Compounds Mohammad Youmf; Diamond Anvil Cells in High Pressure Studies of Semiconductors

Volume 56 Germanium Silicon J. C. Bean, Growth Techniques and Procedures D. E. Savage, F. Liu, V. Zielasek, and M. G. Lagally, Fundamental Crystal Growth Mechanisms R Hull, Misfit Strain Accommodation in SiGe Heterostructures M.J. Shaw and M. Jaros, Fundamental Physics of Strained Layer GeSk Quo Vadis? F. Cerdeira. Optical Properties S. A. Ringel and P. N. Grillot, Electronic Properties and Deep Levels in Germanium-Silicon J. C Campbelf, Optoelectronics in Silicon and Germanium Silicon K Eberl. XI Brunner. and 0.G. Schmidt, Si, -,CY and Si, -=-, Ge,C, Alloy Layers

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Volume 57 Gallium Nitride (GaN) Il Richard J. Molnur, Hydride Vapor Phase Epitaxial Growth of Ill-V Nitrides T. D. Moustukus. Growth of 111-V Nitrides by Molecular Beam Epitaxy Zuzunnu Lilientul- Weber, Defects in Bulk GaN and Homoepitaxial Layers Chris G. Vun de Wulk und Noble M. Johnson. Hydrogen in Ill-V Nitrides W. G6rz und N. M. Johnson, Characterization of Dopants and Deep Level Defects in Gallium Nitride Brrnurd Gill,Stress Effects on Optical Properties Christian Kisielowski. Strain in GaN Thin Films and Heterostructures Joseph A. Mirugliottu and Dennis K Wickenden. Nonlinear Optical Properties of Gallium Nitride 8. K. Meyer, Magnetic Resonance Investigationson Group 111-Nitrides .M. S. Shur und M. Asif Khan. GaN and AIGaN Ultraviolet Detectors C. H. Qiu. J. 1. Punkove. and C. Rossingfon. 111-V Nitride-Based X-ray Detectors

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